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34 MultiView scientific calculator comes with the same features that made the TI-34 II Explorer Plus so helpful at exploring fraction simplification, integer division and constant operators. Enter statistical data for 1- and 2-var analysis as well as for exploring patterns via list conversions to see different number formats like decimal, fraction and percent side-by-side. Quickly view ... TI s 2-line scientific with more advanced features. 1 brand used by students. Easily scroll review and edit current or previous entries. Perform a variety of conversions and advanced calculations. See both equation and answer. Solar and battery powered. Built to last. Slide on case included. Bulk Packaged. Appropriate for:. General Math. Pre-Algebra. Algebra I and II. Geometry. Trigonometry.The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom.Color: Orange
Product Description About the Author Catherine Twomey Fosnot is the Founding Director of Mathematics in the City and former Professor of Education at The City College of the City of New York. She has twice received the "best writing" award from AERA's Constructivist SIG and she was the recipient of the "young scholar" award by Educational Communication and Technology Journal. She is the lead author of the Contexts for Learning Mathematics series as well as the Young Mathematicians at Work series. Bill Jacob is a Professor of Mathematics at the University of California, Santa Barbara. In addition to his mathematical research, he develops and teaches courses for pre-service teachers. He is coauthor with Catherine Fosnot of Young Mathematicians at Work: Constructing Algebra and has been a collaborator with Mathematics in the City for twelve years. This book offers deep insights into what algebra is and how it can be developed from the very youngest grades. Detailed classroom descriptions offer both useful tasks and excellent teaching strategies. True to its title, it aims to develop "young mathematicians" who engage in developing, testing, and proving their own mathematical ideas. I particularly appreciated the attention the authors paid to bridging individual and collective understanding. By the end of the book, Grade 5 students are solving systems of equations!
Skillfully conceived and written text, with many special features, covers functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, patterns for integration, differential equations, much more. Many examples, exercises and practice problems, with answers. Advanced undergraduate/graduate-level. development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss and Riemann is a story that is often broken into parts Ц axiomatic geometry, non-Euclidean geometry and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Exercise Genomics encompasses the translation of exercise genomics into preventive medicine by presenting a broad overview of the rapidly expanding research examining the role of genetics and genomics within the areas of exercise performance and health-related physical activity. 'Iron Claw: Beats, Glitches, Bass & Leads' is an ultimate collection of 100 WAV samples recorded at 90 BPM for multimedia projects. Geared with the most recent technology, this new Temporal Geometry sample pack is a massive library of cutting-edge audio elements you can use right out of the box within your projects. As these samples are all royalty-free, you can use them into your music production, game or TV project without having to worry about any additional licencing fees.
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1992, I had the honor and the pleasure of working with numerous students and realized the need for prep books that can simply explain the fundamentals of mathematics. This book is built on many years of research and experience in this field. Most importantly, the questions in this book focus on building a solid understanding of basic mathematical concepts. Without understanding these solid foundations, it will be difficult to score well on these exams. This book emphasizes that any difficult math question can all be solved with a solid understanding of basic concepts. "Perfect Tips and 8 full length practice tests (4 AP Calculus AB and 4 AP Calculus BC)"
College Algebra Our popular College Algebra course provides a working knowledge of college-level algebra and its applications. Emphasis is placed upon the solution and the application of linear and quadratic equations, word problems, polynomials, and rational and radical equations. Students perform operations on real numbers and polynomials and simplify algebraic, rational, and radical expressions.
Graw-Hill Education Announces Interactive E-books with ALEKS 360 McGraw-Hill Education has introduced ALEKS 360, a mathematics solution that combines an artificial intelligence and personalized learning program with a fully integrated, interactive e-book package. ALEKS 360 delivers assessments of students' math knowledge, guiding them in the selection of appropriate new study material, and recording their progress toward mastery of course goals. Through adaptive questioning, ALEKS accurately assesses a student's knowledge state and delivers targeted instruction on the exact topics a student is most ready to learn. The e-books featured within ALEKS 360 are interactive versions of their physical counterparts, which offer virtual features such as highlighting and note-taking capabilities, as well as access to multimedia assets such as images, video, and homework exercises. E-books are accessible from ALEKS Student Accounts and the ALEKS Instructor Module for convenient, direct access. The initial e-books to be offered in ALEKS 360 include: Introductory Algebra, Second Edition, by Julie Miller and Molly O'Neill; Intermediate Algebra, Second Edition, by Miller and O'Neill; College Precalculus, Second Edition, by John W. Coburn; and College Algebra, Second Edition, by Coburn.
978047113 of the Quintic A chance for students to apply a wide range of mathematics to an engaging problem This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned. The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem--solving the quintic. The problem is approached from two directions: the first is Felix Klein's nineteenth-century approach, using the icosahedron. The second approach presents recent works of Peter Doyle and Curt McMullen, which update Klein's use of transcendental functions to a solution through pure iteration. Filling a pedagogical gap in the literature and providing a solid platform from which to address more advanced material, this meticulously written book: * Develops the Riemann sphere and its field of functions, classifies the finite groups of its automorphisms, computes for each such group a generator of the group-invariant functions, and discusses algebraic aspects of inverting this generator * Gives, in the case of the icosahedral group, an elegant presentation of the relevant icosahedral geometry and its relation to the Brioschi quintic * Reduces the general quintic to Brioschi form by radicals * Proves Kronecker's theorem that an "auxiliary" square root is necessary for any such reduction * Expounds Doyle and McMullen's development of an iterative solution to the quintic * Provides a wealth of exercises and illustrations to clarify the geometry of the quintic
This course includes lecture notes, assignments, problems for group work in recitation, and a full set of lecture videos.... see more This course includes lecture notes, assignments, problems for group work in recitation, and a full set of lecture videos. These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring 2003. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectur This archive is designed as a resource for enriching your courses with mathematical Fun Facts! It is designed to pique the... see more This archive is designed as a resource for enriching your courses with mathematical Fun Facts! It is designed to pique the interest of students in different areas of mathematics. The fun facts were originally conceived as five minute warm ups at the beginning of lectures so that non mathematics majors would not think math was just calculus. Presentation suggestions are also givenOver the past year, more than 50 undergraduates from around the US havecollaborated to establish the National Journal of... see more Over the past year, more than 50 undergraduates from around the US havecollaborated to establish the National Journal of Young Investigators (JYI), "״a faculty and student reviewed, peer edited and published, national journal״" of science and engineering. Designed to showcase undergraduate research and serve as a hub for up-and-coming scientists, JYI's staff is composed entirely of undergraduate students from academic institutions across the US. The journal's scientific articles are organized in three subject areas: Physical Sciences and Mathematics, Biological and Biomedical Sciences, and Basic Engineering Sciences. The full text of current articles may be viewed online, and interested contributors will find instructions for article submissions on-site." Develops a working knowledge of and ability to apply numerical methods in solving some basic mathematical problems such as... see more Develops a working knowledge of and ability to apply numerical methods in solving some basic mathematical problems such as interpolation, numerical integration, and finding roots of functions.Compulsory Readings for Numerical Methods (PDF) The Polar Curves Applet behaves much like a graphing calculator but adds two features: the user has options to (1) plot... see more The Polar Curves Applet behaves much like a graphing calculator but adds two features: the user has options to (1) plot points on the user-specified polar curve given by r(t) beginning at a user-specified value of t; and (2) fill in the area bounded by a user-specified polar curve given by r(t) beginning at a user-specified value of t. This applet was developed for the Mathematical Activities with Java workshop. It is recommended for introducing curves in polar coordinates and for discussing the area bounded by one polar curve. This site is part of an online course in trigonometry taught in the Mathematics Department at the University of Idaho. ... see more This site is part of an online course in trigonometry taught in the Mathematics Department at the University of Idaho. Contents include approximately fifty streaming videos (RealPlayer) of classroom-type instruction on topics ranging from "The Cartesian Coordiate System" to "Inverse Trigonometic Functions." A downloadable problem set with odd-number solutions is also available. Probability has applications in classical physics, but it plays a more essential role in quantum physics. We can take... see more Probability has applications in classical physics, but it plays a more essential role in quantum physics. We can take examples from quantum mechanics without raising the mathematical or computational level appreciably, but for those with no previous knowledge of the subject, the brief outline given below may not be adequate. For my technology class I was asked to complete a StaIR. This stands for Stand-Alone Instructional Resource. This is a... see more For my technology class I was asked to complete a StaIR. This stands for Stand-Alone Instructional Resource. This is a resource that a student could use to learn a topic completely independently. It teaches the topic, includes independent practice, and feedback. When asked to do this I wanted to make this for something that could be used within my classroom. I chose to complete a resource for the Pythagorean Theorem. This is a standard that is necessary for all seventh grade students. I will be able to use this to differentiate this in both my Math 7 and Pre-Algebra class. A presentation on the nature of science and how it relates to quantum mechanics. Level is for high school to beginning... see more A presentation on the nature of science and how it relates to quantum mechanics. Level is for high school to beginning college. Takes the viewer through the quirks and seeming paradoxes of quantum mechanics, including the double slit experiment and Schrodinger's Cat.
Oftentimes, when it comes to math, some people struggle with the concepts, finding it too difficult. But follow these few steps and Math will be easy in every class. Ad Steps 1 Master Your Basics: The number one reason that people struggle in math class is because their basics and their fundamentals are not fully developed. Algebra and Geometry are the building blocks for the more advanced math later on (Calculus, Differential Equations, etc...). Ad 2 Get Ahead: Most schools give you a textbook for math and it's a pretty big book. What you can do is, study ahead. Whenever you have time, you can look a section ahead, and be prepared for tomorrow's material. 3 Self-Study: This is the most efficient way of studying math. I would recommend you to buy math textbooks from a local bookstore. You can also search on the internet for great math books.Don't get a book that is very short (100 pages) for a topic like Geometry. Get a textbook or a few workbooks on the topic. It's good to buy more than one book, since some books leave out certain things. 4 Studying: When you self-study, it's good to have the book and a notebook with you, college ruled preferably. Write down all the vocabulary and terms and the example problems. You don't have to do each and every single practice problem if you find it repetitive, just have an intuitive answer. (As long as you know the process of solving it) It's also good to get into a habit of working on more Word Problems, which can help you apply the concept into real-life situations. 5 Competition: If you do enough self-studying, and you look through your studying notes when you have free time, you should already have a very good basis in math. If you're a fast learner, then it would be even better since you can learn the higher level concepts quicker. If your school has a math club or team that you can join, go for it! Chances are, you'll meet individuals who are very talented in math and can help you expand your knowledge by attending competitions. 6 Loving Math: Once you do this part, math would be no challenge whatsoever. Once you get good in math, help others, it's okay to show off you knowledge, in a good way. Once you start to take interest in math and start studying it and attend math competitions and expand your knowledge on math, you will love it. Once you have a passion for math, you will want to learn more, achieve more, and become the mathematician you've never imagined
This marvellous book, dedicated to Dick Askey with a foreword by Mizan Rahman, gives a very nice introduction to the new branch of classical analysis called Basic Fourier Series. The author is a great expert in this new field and the book is much more than just an introduction to q-Fourier Analysis. The book comprises, besides the foreword and a preface, twelve chapters, six appendices, a quite extensive bibliography and an index. Each chapter, except for the last one, concludes with a nice collection of exercises, which makes it suitable for a course on the subject. The first two chapters form a quite elementary introduction to basic exponential and trigonometric functions. Chapter 3 deals with addition theorems for these functions and in Chapter 4 the expansions in terms of these functions are discussed. The fifth chapter is an introduction to basic Fourier series, followed by a thorough investigation in Chapter 6. The latter chapter deals with, for instance, asymptotics of zeros, methods of summation and analytic continuation of basic Fourier series. Chapter 7 deals with completeness of basic trigonometric systems in general, illustrated by some important examples. Chapter 8 studies the asymptotics of zeros in more detail, which leads to improved results in comparison with the asymptotics obtained in Chapter 6. Chapter 9 deals with expansions in basic Fourier series; many explicit expansions are given of (basic) elementary special functions. In Chapter 10 the author introduces basic Bernoulli and Euler polynomials and numbers, and also a basic extension of the Riemann zeta function. Chapter 11 deals with a numerical investigation of basic Fourier series and in Chapter 12 the author gives several suggestions for further work in the theory of q-Fourier Analysis and related topics. The appendices comprise a selection of basic summation and transformation formulas, some theorems of complex analysis, tables of zeros of basic sine and cosine functions and some numerical examples.
Elementary Number Theory Review: Elementary Number Theory (AMS Chelsea Publishing) User Review - Pietro - Goodreads For an "elementary" textbook, contains some pretty interesting "intermediate" topics. Includes an elementary proof of the prime number theorem (but instead of the tight result of Hadamard and Poussin, gets a range like 3/4 < pi(x)*log(x)/x < 5/4).Read full review
2 At4 Both simple and accessible, Math in Minutes is a visually led introduction to 200 key mathematical concepts. Each concept is described by means of an easy-to-understand illustration and a compact, 200-word explanation.
Yes, the course is fast-paced, and it will be difficult to catch up with some aspects of math (such as trigonometry) during the semester.You need your focus to master the problem-solving that physics challenges you to do. Yes, it often does help to have had physics in high school before. But it is no guarantee for success, nor do you need to worry about failure if you did not have physics before. There is really no clear correlation. And no – even with physics in high school I guarantee you that you will be challenged. If you are concerned whether you are prepared enough before the course, below are some things you can do to put yourself into a good position before the semester. They are arranged in order of decreasing significance, with the last one still being pretty significant. You may also refer to appendix B in your text book to see examples of what mathematically lies ahead of you. 1.Review trigonometry and geometry! Mastering these is more important for your comfort in this class than remembering some calculus. Most of the semester, we will spend representing all sorts of physical phenomena using vectors. Those little arrow symbols combine in triangles and polygons on the paper. Once you have drawn them, you will use the drawings to calculate unknown lengths, angles, etc. which means: there is geometry and trigonometry everywhere. Often we need to draw from every trick in the math book to solve the problem. d.Big one: be able to solve a system of equations for multiple variables. In this semster, we will usually have two or three equations and two or three variables. Mathematics has equipped you with several tools to deal with a system of equations, the simplest one being repeated substitution. 3.Review a few rules for integration and differentiation! We will use differentiation and integration in some well-defined places, and use mostly very simple examples. The lecture and the text book will introduce these places in a way similar to what you may have seen in your calculus course. However, after the introduction, you will sometimes be confronted just to "do" a derivative or integral of some function, and for that it is useful to remember a few simple rules. 4.Enjoy reading a few physics-related texts other than the text book– and that one is really up to you. You may wonder about the rules governing the motion of satellites, or why your tires spin on snow (easier the more you push the gas), or why your bike does not tip over when it is moving. Why is this penny speeding up on its way down the wishing well? Why do the Australians not hang upside down, and how does this go together with our image of "gravity constant and down"? Why is there a gap between the top of the steam vent and the white cloud, and why is it rising? Where will this birthday balloon land if the wind from the west is increasing in speed and turning SW with height above the ground? Look around and wonder about things that you are taking for granted every day, and bring those thoughts to class …
Why Take Math in High School Do you want to travel to Mars? Design houses or create computer software? Do you want to discover a cure to cancer or protect rivers and oceans from pollution? If you do, be sure to take lots of math in school. Many challenging and rewarding careers - not just engineering - demand a strong background in math. Classes like algebra, geometry, and trigonometry may be difficult, but they will open the doors to many exciting opportunities in your future. Of course, you can catch up on your math when you get to college, but like a language it is much easier to really learn it when you are young, and you'll have a hard time catching up in college without taking extra years if you don't at least have the basics. Since you have to take most math class in a particular order, it's also important to start early so that you are prepared to take the right classes when you need them. Besides the inherent beauty in mathematics there are lots of other reasons to take as much math as you can, even if you don't want to be an engineer. For one, math can actually make you smarter! It's like endurance training for your brain. Learning to think critically and focus on a problem is important for any career. You may also want to take math so you can make and save more money. On average, people who understand math have higher paying careers. That may not always be the case, but you'll certainly increase your odds by understanding as much as you can. And when all your friends are losing their last dime on the latest crazy scheme that doesn't make mathematical sense, you might actually have the forethought to sit back and crunch the numbers before jumping in. Having a solid background in math is very much like understanding a very useful and universal language. It is important to start early to be completely fluent, and you'll be able to communicate difficult concepts with a few simple equations. People in many other countries know the importance of math and work very hard to master it from an early age. Considering the increasing globalization of the economy, you shouldn't be surprised that you will be competing with many of those hard-working students for jobs by the time you get out of college. And speaking of college, you may want to consider taking extra math just to help make sure you get into the right college. A compelling essay will go a long way on your college entrance applications, but so will a solid background in mathematics. And once you get to college, you won't have to take loads of remedial math courses just to catch up. And finally, you should take math because you're cool! You won't need to explain away why you don't do numbers for the rest of your life. You'll be able to help your kids with their math homework, and you will be the one people turn to when they need some creative problem-solving. So now that you know all the great reasons to study math in high school, here are some math classes that you will want to take: Algebra Algebra is extremely useful for solving problems. Algebra uses basic arithmetic rules to describe and group things and to discover the value of something unknown (usually represented by a letter in an equation). Algebra is the foundation for many other math subjects. Geometry Geometry is the study of the properties of and relationships between points, lines, angles, and surfaces. Geometry uses logic and mathematical laws to describe the physical world and will give you several other important problem solving tools. Trigonometry In trigonometry, you study triangles and trigonometric functions like sine, cosine, and tangent. Trigonometry has real world applications dealing with everything from radio waves and electricity to telescopes and ship navigation. Calculus With calculus, you combine everything you've learned about math and take the next step. Calculus uses special symbols and logic to do difficult calculations, like determining the orbit of a space vehicle, or predicting the time it takes a car to stop on a wet road. Calculus is a very powerful tool for solving complex problems. What about you? So do you have a favorite math class? Can you think of other great reasons for taking math? Tell us in the comments below. I love math! I understood and understand why math is so important now in life. I want to go far in life and be a pediatric neurosurgeon and I know it would take alot. I realize math is important in life and I should continue to study math. I really love math and now I know that it can really help me get more opportunities for career choices. Even though it is my favorite subject I still need to pay close attention and take all the math classes that I can. I really strong but math is not my forte. After reading this article i realize now that math is necessary.Even though i don't like it but i realize in the distant future i may be able to enjoy math. Thank you for this informative read i am pretty weak at math like in 60s. this is because i ignore math when i was in elementary school. now i really want math to become an engineer. how can i learn basic math that i have missed in one year. pls help!!!!!!
IntroductionPurpose of this documentThis publication is intended to guide the planning, teaching and assessment of the subject in schools. Subjectteachers are the primary audience, although it is expected that teachers will use the guide to inform studentsand parents about the subject.This guide can be found on the subject page of the online curriculum centre (OCC) at apassword-protected IB website designed to support IB teachers. It can also be purchased from the IB store at resourcesAdditional publications such as teacher support materials, subject reports, internal assessment guidanceand grade descriptors can also be found on the OCC. Specimen and past examination papers as well asmarkschemes can be purchased from the IB store.Teachers are encouraged to check the OCC for additional resources created or used by other teachers. Teacherscan provide details of useful resources, for example: websites, books, videos, journals or teaching ideas. First examinations 2014 Mathematics HL guide 1 IntroductionThe Diploma ProgrammeThe Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable andinquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to developintercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate arange of points of view.The Diploma Programme hexagonThe course is presented as six academic areas enclosing a central core (see figure 1). It encourages the concurrentstudy of a broad range of academic areas. Students study: two modern languages (or a modern language anda classical language); a humanities or social science subject; an experimental science; mathematics; one ofthe creative arts. It is this comprehensive range of subjects that makes the Diploma Programme a demandingcourse of study designed to prepare students effectively for university entrance. In each of the academic areasstudents have flexibility in making their choices, which means they can choose subjects that particularlyinterest them and that they may wish to study further at university. Studies in language and literature Group 1 Language Individuals acquisition and societies Group 2 ledg e ext Group 3 now e LEARN nd k IB of ed E theory ER essay TH PR OFIL E cr ea tivi ice Experimental Group 4 ty, action, serv Group 5 Mathematics sciences Group 6 The arts Figure 1 Diploma Programme model2 Mathematics HL guide The Diploma ProgrammeChoosing the right combinationStudents are required to choose one subject from each of the six academic areas, although they can choose asecond subject from groups 1 to 5 instead of a group 6 subject. Normally, three subjects (and not more thanfour) are taken at higher level (HL), and the others are taken at standard level (SL). The IB recommends 240teaching hours for HL subjects and 150 hours for SL. Subjects at HL are studied in greater depth and breadththan at SL.At both levels, many skills are developed, especially those of critical thinking and analysis. At the end ofthe course, students' abilities are measured by means of external assessment. Many subjects contain someelement of coursework assessed by teachers. The courses are available for examinations in English, French andSpanish, with the exception of groups 1 and 2 courses where examinations are in the language of study.The core of the hexagonAll Diploma Programme students participate in the three course requirements that make up the core of thehexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the DiplomaProgramme.The theory of knowledge course encourages students to think about the nature of knowledge, to reflect onthe process of learning in all the subjects they study as part of their Diploma Programme course, and to makeconnections across the academic areas. The extended essay, a substantial piece of writing of up to 4,000 words,enables students to investigate a topic of special interest that they have chosen themselves. It also encouragesthem to develop the skills of independent research that will be expected at university. Creativity, action, serviceinvolves students in experiential learning through a range of artistic, sporting, physical and service activities.The IB mission statement and the IB learner profileThe Diploma Programme aims to develop in students the knowledge, skills and attitudes they will need tofulfill the aims of the IB, as expressed in the organization's mission statement and the learner profile. Teachingand learning in the Diploma Programme represent the reality in daily practice of the organization's educationalphilosophy. Mathematics HL guide 3 IntroductionNature of the subjectIntroductionThe nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probablya combination of these, but there is no doubt that mathematical knowledge provides an important key tounderstanding the world in which we live. Mathematics can enter our lives in a number of ways: we buyproduce in the market, consult a timetable, read a newspaper, time a process or estimate a length. Mathematics,for most of us, also extends into our chosen profession: visual artists need to learn about perspective; musiciansneed to appreciate the mathematical relationships within and between different rhythms; economists needto recognize trends in financial dealings; and engineers need to take account of stress patterns in physicalmaterials. Scientists view mathematics as a language that is central to our understanding of events that occurin the natural world. Some people enjoy the challenges offered by the logical methods of mathematics andthe adventure in reason that mathematical proof has to offer. Others appreciate mathematics as an aestheticexperience or even as a cornerstone of philosophy. This prevalence of mathematics in our lives, with all itsinterdisciplinary connections, provides a clear and sufficient rationale for making the study of this subjectcompulsory for students studying the full diploma.Summary of courses availableBecause individual students have different needs, interests and abilities, there are four different courses inmathematics. These courses are designed for different types of students: those who wish to study mathematicsin depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; thosewho wish to gain a degree of understanding and competence to understand better their approach to othersubjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in theirdaily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great careshould be taken to select the course that is most appropriate for an individual student.In making this selection, individual students should be advised to take account of the following factors:• their own abilities in mathematics and the type of mathematics in which they can be successful• their own interest in mathematics and those particular areas of the subject that may hold the most interest for them• their other choices of subjects within the framework of the Diploma Programme• their academic plans, in particular the subjects they wish to study in future• their choice of career.Teachers are expected to assist with the selection process and to offer advice to students.4 Mathematics HL guide Nature of the subjectMathematical studies SLThis course is available only at standard level, and is equivalent in status to mathematics SL, but addressesdifferent needs. It has an emphasis on applications of mathematics, and the largest section is on statisticaltechniques. It is designed for students with varied mathematical backgrounds and abilities. It offers studentsopportunities to learn important concepts and techniques and to gain an understanding of a wide varietyof mathematical topics. It prepares students to be able to solve problems in a variety of settings, to developmore sophisticated mathematical reasoning and to enhance their critical thinking. The individual project is anextended piece of work based on personal research involving the collection, analysis and evaluation of data.Students taking this course are well prepared for a career in social sciences, humanities, languages or arts.These students may need to utilize the statistics and logical reasoning that they have learned as part of themathematical studies SL course in their future studies.Mathematics SLThis course caters for students who already possess knowledge of basic mathematical concepts, and who areequipped with the skills needed to apply simple mathematical techniques correctly. The majority of thesestudents will expect to need a sound mathematical background as they prepare for future studies in subjectssuch as chemistry, economics, psychology and business administration.Mathematics HLThis course caters for students with a good background in mathematics who are competent in a range ofanalytical and technical skills. The majority of these students will be expecting to include mathematics asa major component of their university studies, either as a subject in its own right or within courses such asphysics, engineering and technology. Others may take this subject because they have a strong interest inmathematics and enjoy meeting its challenges and engaging with its problems.Further mathematics HLThis course is available only at higher level. It caters for students with a very strong background in mathematicswho have attained a high degree of competence in a range of analytical and technical skills, and who displayconsiderable interest in the subject. Most of these students will expect to study mathematics at university, eitheras a subject in its own right or as a major component of a related subject. The course is designed specificallyto allow students to learn about a variety of branches of mathematics in depth and also to appreciate practicalapplications. It is expected that students taking this course will also be taking mathematics HL.Note: Mathematics HL is an ideal course for students expecting to include mathematics as a major componentof their university studies, either as a subject in its own right or within courses such as physics, engineeringor technology. It should not be regarded as necessary for such students to study further mathematics HL.Rather, further mathematics HL is an optional course for students with a particular aptitude and interest inmathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means anecessary qualification to study for a degree in mathematics.Mathematics HL—course detailsThe course focuses on developing important mathematical concepts in a comprehensible, coherent andrigorous way. This is achieved by means of a carefully balanced approach. Students are encouraged to applytheir mathematical knowledge to solve problems set in a variety of meaningful contexts. Development of eachtopic should feature justification and proof of results. Students embarking on this course should expect todevelop insight into mathematical form and structure, and should be intellectually equipped to appreciate thelinks between concepts in different topic areas. They should also be encouraged to develop the skills needed tocontinue their mathematical growth in other learning environments. Mathematics HL guide 5 Nature of the subjectThe internally assessed component, the exploration, offers students the opportunity for developingindependence in their mathematical learning. Students are encouraged to take a considered approach tovarious mathematical activities and to explore different mathematical ideas. The exploration also allowsstudents to work without the time constraints of a written examination and to develop the skills they need forcommunicating mathematical ideas.This course is a demanding one, requiring students to study a broad range of mathematical topics througha number of different approaches and to varying degrees of depth. Students wishing to study mathematicsin a less rigorous environment should therefore opt for one of the standard level courses, mathematics SL ormathematical studies SL. Students who wish to study an even more rigorous and demanding course shouldconsider taking further mathematics HL in addition to mathematics HL.Prior learningMathematics is a linear subject, and it is expected that most students embarking on a Diploma Programme(DP) mathematics course will have studied mathematics for at least 10 years. There will be a great varietyof topics studied, and differing approaches to teaching and learning. Thus students will have a wide varietyof skills and knowledge when they start the mathematics HL course. Most will have some background inarithmetic, algebra, geometry, trigonometry, probability and statistics. Some will be familiar with an inquiryapproach, and may have had an opportunity to complete an extended piece of work in mathematics.At the beginning of the syllabus section there is a list of topics that are considered to be prior learning for themathematics HL course. It is recognized that this may contain topics that are unfamiliar to some students, but itis anticipated that there may be other topics in the syllabus itself that these students have already encountered.Teachers should plan their teaching to incorporate topics mentioned that are unfamiliar to their students.Links to the Middle Years ProgrammeThe prior learning topics for the DP courses have been written in conjunction with the Middle YearsProgramme (MYP) mathematics guide. The approaches to teaching and learning for DP mathematics buildon the approaches used in the MYP. These include investigations, exploration and a variety of differentassessment tools.A continuum document called Mathematics: The MYP–DP continuum (November 2010) is available on theDP mathematics home pages of the OCC. This extensive publication focuses on the alignment of mathematicsacross the MYP and the DP. It was developed in response to feedback provided by IB World Schools, whichexpressed the need to articulate the transition of mathematics from the MYP to the DP. The publication alsohighlights the similarities and differences between MYP and DP mathematics, and is a valuable resource forteachers.Mathematics and theory of knowledgeThe Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed thatthese all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by datafrom sense perception, mathematics is dominated by reason, and some mathematicians argue that their subjectis a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceivebeauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge.6 Mathematics HL guide Nature of the subjectAs an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. Thismay be related to the "purity" of the subject that makes it sometimes seem divorced from reality. However,mathematics has also provided important knowledge about the world, and the use of mathematics in scienceand technology has been one of the driving forces for scientific advances.Despite all its undoubted power for understanding and change, mathematics is in the end a puzzlingphenomenon. A fundamental question for all knowers is whether mathematical knowledge really existsindependently of our thinking about it. Is it there "waiting to be discovered" or is it a human creation?Students' attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, andthey should be encouraged to raise such questions themselves, in mathematics and TOK classes. This includesquestioning all the claims made above. Examples of issues relating to TOK are given in the "Links" column ofthe syllabus. Teachers could also discuss questions such as those raised in the "Areas of knowledge" section ofthe TOK guide.Mathematics and the international dimensionMathematics is in a sense an international language, and, apart from slightly differing notation, mathematiciansfrom around the world can communicate within their field. Mathematics transcends politics, religion andnationality, yet throughout history great civilizations owe their success in part to their mathematicians beingable to create and maintain complex social and architectural structures.Despite recent advances in the development of information and communication technologies, the globalexchange of mathematical information and ideas is not a new phenomenon and has been essential to theprogress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuriesago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websitesto show the contributions of different civilizations to mathematics, but not just for their mathematical content.Illustrating the characters and personalities of the mathematicians concerned and the historical context inwhich they worked brings home the human and cultural dimension of mathematics.The importance of science and technology in the everyday world is clear, but the vital role of mathematicsis not so well recognized. It is the language of science, and underpins most developments in science andtechnology. A good example of this is the digital revolution, which is transforming the world, as it is all basedon the binary number system in mathematics.Many international bodies now exist to promote mathematics. Students are encouraged to access the extensivewebsites of international mathematical organizations to enhance their appreciation of the internationaldimension and to engage in the global issues surrounding the subject.Examples of global issues relating to international-mindedness (Int) are given in the "Links" column of thesyllabus. Mathematics HL guide 7 IntroductionAimsGroup 5 aimsThe aims of all mathematics courses in group 5 are to enable students to:1. enjoy mathematics, and develop an appreciation of the elegance and power of mathematics2. develop an understanding of the principles and nature of mathematics3. communicate clearly and confidently in a variety of contexts4. develop logical, critical and creative thinking, and patience and persistence in problem-solving5. employ and refine their powers of abstraction and generalization6. apply and transfer skills to alternative situations, to other areas of knowledge and to future developments7. appreciate how developments in technology and mathematics have influenced each other8. appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics9. appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives10. appreciate the contribution of mathematics to other disciplines, and as a particular "area of knowledge" in the TOK course.8 Mathematics HL guide IntroductionAssessment objectivesProblem-solving is central to learning mathematics and involves the acquisition of mathematical skills andconcepts in a wide range of situations, including non-routine, open-ended and real-world problems. Havingfollowed a DP mathematics HL course, students will be expected to demonstrate the following.1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.3. Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.4. Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.5. Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions.6. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions and testing their validity. Mathematics HL guide 9 SyllabusSyllabus outline Teaching hours Syllabus component HL All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as prior learning. Topic 1 30 Algebra Topic 2 22 Functions and equations Topic 3 22 Circular functions and trigonometry Topic 4 24 Vectors Topic 5 36 Statistics and probability Topic 6 48 Calculus Option syllabus content 48 Students must study all the sub-topics in one of the following options as listed in the syllabus details. Topic 7 Statistics and probability Topic 8 Sets, relations and groups Topic 9 Calculus Topic 10 Discrete mathematics Mathematical exploration 10 Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. Total teaching hours 24010 Mathematics HL guide SyllabusApproaches to the teaching and learningof mathematics HLThroughout the DP mathematics HL course, students should be encouraged to develop their understandingof the methodology and practice of the discipline of mathematics. The processes of mathematical inquiry,mathematical modelling and applications and the use of technology should be introduced appropriately.These processes should be used throughout the course, and not treated in isolation.Mathematical inquiryThe IB learner profile encourages learning by experimentation, questioning and discovery. In the IBclassroom, students should generally learn mathematics by being active participants in learning activitiesrather than recipients of instruction. Teachers should therefore provide students with opportunities to learnthrough mathematical inquiry. This approach is illustrated in figure 2. Explore the context Make a conjecture Test the conjecture Reject Accept Prove Extend Figure 2 Mathematics HL guide 11 Approaches to the teaching and learning of mathematics HLMathematical modelling and applicationsStudents should be able to use mathematics to solve problems in the real world. Engaging students in themathematical modelling process provides such opportunities. Students should develop, apply and criticallyanalyse models. This approach is illustrated in figure 3. Pose a real-world problem Develop a model Test the model Reject Accept Reflect on and apply the model Extend Figure 3TechnologyTechnology is a powerful tool in the teaching and learning of mathematics. Technology can be used to enhancevisualization and support student understanding of mathematical concepts. It can assist in the collection,recording, organization and analysis of data. Technology can increase the scope of the problem situations thatare accessible to students. The use of technology increases the feasibility of students working with interestingproblem contexts where students reflect, reason, solve problems and make decisions.As teachers tie together the unifying themes of mathematical inquiry, mathematical modelling andapplications and the use of technology, they should begin by providing substantial guidance, and thengradually encourage students to become more independent as inquirers and thinkers. IB students should learnto become strong communicators through the language of mathematics. Teachers should create a safe learningenvironment in which students are comfortable as risk-takers.Teachers are encouraged to relate the mathematics being studied to other subjects and to the real world,especially topics that have particular relevance or are of interest to their students. Everyday problems andquestions should be drawn into the lessons to motivate students and keep the material relevant; suggestionsare provided in the "Links" column of the syllabus. The mathematical exploration offers an opportunityto investigate the usefulness, relevance and occurrence of mathematics in the real world and will add anextra dimension to the course. The emphasis is on communication by means of mathematical forms (for12 Mathematics HL guide Approaches to the teaching and learning of mathematics HLexample, formulae, diagrams, graphs and so on) with accompanying commentary. Modelling, investigation,reflection, personal engagement and mathematical communication should therefore feature prominently in theDP mathematics classroom.For further information on "Approaches to teaching a DP course", please refer to the publication The DiplomaProgramme: From principles into practice (April 2009). To support teachers, a variety of resources can befound on the OCC and details of workshops for professional development are available on the public website.Format of the syllabus• Content: this column lists, under each topic, the sub-topics to be covered.• Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.• Links: this column provides useful links to the aims of the mathematics HL course, with suggestions for discussion, real-life examples and ideas for further investigation. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows. Appl real-life examples and links to other DP subjects Aim 8 moral, social and ethical implications of the sub-topic Int international-mindedness TOK suggestions for discussionNote that any syllabus references to other subject guides given in the "Links" column are correct for thecurrent (2012) published versions of the guides.Notes on the syllabus• Formulae are only included in this document where there may be some ambiguity. All formulae required for the course are in the mathematics HL and further mathematics HL formula booklet.• The term "technology" is used for any form of calculator or computer that may be available. However, there will be restrictions on which technology may be used in examinations, which will be noted in relevant documents.• The terms "analysis" and "analytic approach" are generally used when referring to an approach that does not use technology.Course of studyThe content of all six topics and one of the option topics in the syllabus must be taught, although not necessarilyin the order in which they appear in this guide. Teachers are expected to construct a course of study thataddresses the needs of their students and includes, where necessary, the topics noted in prior learning. Mathematics HL guide 13 Approaches to the teaching and learning of mathematics HLIntegration of the mathematical explorationWork leading to the completion of the exploration should be integrated into the course of study. Details of howto do this are given in the section on internal assessment and in the teacher support material.Time allocationThe recommended teaching time for higher level courses is 240 hours. For mathematics HL, it is expected that10 hours will be spent on work for the exploration. The time allocations given in this guide are approximate,and are intended to suggest how the remaining 230 hours allowed for the teaching of the syllabus mightbe allocated. However, the exact time spent on each topic depends on a number of factors, including thebackground knowledge and level of preparedness of each student. Teachers should therefore adjust thesetimings to correspond to the needs of their students.Use of calculatorsStudents are expected to have access to a graphic display calculator (GDC) at all times during the course.The minimum requirements are reviewed as technology advances, and updated information will be providedto schools. It is expected that teachers and schools monitor calculator use with reference to the calculatorpolicy. Regulations covering the types of calculators allowed in examinations are provided in the Handbook ofprocedures for the Diploma Programme. Further information and advice is provided in the Mathematics HL/SL: Graphic display calculators teacher support material (May 2005) and on the OCC.Mathematics HL and further mathematicsHL formula bookletEach student is required to have access to a clean copy of this booklet during the examination. It isrecommended that teachers ensure students are familiar with the contents of this document from the beginningof the course. It is the responsibility of the school to download a copy from IBIS or the OCC, check that thereare no printing errors, and ensure that there are sufficient copies available for all students.Teacher support materialsA variety of teacher support materials will accompany this guide. These materials will include guidance forteachers on the introduction, planning and marking of the exploration, and specimen examination papers andmarkschemes.Command terms and notation listTeachers and students need to be familiar with the IB notation and the command terms, as these will be usedwithout explanation in the examination papers. The "Glossary of command terms" and "Notation list" appearas appendices in this guide.14 Mathematics HL guide SyllabusPrior learning topicsAs noted in the previous section on prior learning, it is expected that all students have extensive previousmathematical experiences, but these will vary. It is expected that mathematics HL students will be familiar with thefollowing topics before they take the examinations, because questions assume knowledge of them. Teachers musttherefore ensure that any topics listed here that are unknown to their students at the start of the course are included atan early stage. They should also take into account the existing mathematical knowledge of their students to design anappropriate course of study for mathematics HL. This table lists the knowledge, together with the syllabus content,that is essential to successful completion of the mathematics HL course.Students must be familiar with SI (Système International) units of length, mass and time, and their derived units. Topic Content Number Routine use of addition, subtraction, multiplication and division, using integers, decimals and fractions, including order of operations. Rational exponents. Simplification of expressions involving roots (surds or radicals), including rationalizing the denominator. Prime numbers and factors (divisors), including greatest common divisors and least common multiples. Simple applications of ratio, percentage and proportion, linked to similarity. Definition and elementary treatment of absolute value (modulus), a . Rounding, decimal approximations and significant figures, including appreciation of errors. k Expression of numbers in standard form (scientific notation), that is, a × 10 , 1 ≤ a < 10 , k ∈ . Sets and numbers Concept and notation of sets, elements, universal (reference) set, empty (null) set, complement, subset, equality of sets, disjoint sets. Operations on sets: union and intersection. Commutative, associative and distributive properties. Venn diagrams. Number systems: natural numbers; integers, ; rationals, , and irrationals; real numbers, . Intervals on the real number line using set notation and using inequalities. Expressing the solution set of a linear inequality on the number line and in set notation. Mappings of the elements of one set to another; sets of ordered pairs. Mathematics HL guide 15 Prior learning topics Topic Content Algebra Manipulation of linear and quadratic expressions, including factorization, expansion, completing the square and use of the formula. Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included. Linear functions, their graphs, gradients and y-intercepts. Addition and subtraction of simple algebraic fractions. The properties of order relations: <, ≤ , >, ≥ . Solution of linear equations and inequalities in one variable, including cases with rational coefficients. Solution of quadratic equations and inequalities, using factorization and completing the square. Solution of simultaneous linear equations in two variables. Trigonometry Angle measurement in degrees. Compass directions. Right-angle trigonometry. Simple applications for solving triangles. Pythagoras' theorem and its converse. Geometry Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and similarity, including the concept of scale factor of an enlargement. The circle, its centre and radius, area and circumference. The terms arc, sector, chord, tangent and segment. Perimeter and area of plane figures. Properties of triangles and quadrilaterals, including parallelograms, rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes. Volumes of cuboids, pyramids, spheres, cylinders and cones. Classification of prisms and pyramids, including tetrahedra. Coordinate Elementary geometry of the plane, including the concepts of dimension for point, line, plane geometry and space. The equation of a line in the form = mx + c . Parallel and perpendicular lines, y including m1 = m2 and m1m2 = −1 . The Cartesian plane: ordered pairs ( x, y ) , origin, axes. Mid-point of a line segment and distance between two points in the Cartesian plane. Statistics and Descriptive statistics: collection of raw data, display of data in pictorial and diagrammatic probability forms, including frequency histograms, cumulative frequency graphs. Obtaining simple statistics from discrete and continuous data, including mean, median, mode, quartiles, range, interquartile range and percentiles. Calculating probabilities of simple events.16 Mathematics HL guide Syllabus Syllabus content Topic 1—Core: Algebra 30 hoursMathematics HL guide The aim of this topic is to introduce students to some basic algebraic concepts and applications. Content Further guidance Links 1.1 Arithmetic sequences and series; sum of finite Sequences can be generated and displayed in Int: The chess legend (Sissa ibn Dahir). arithmetic series; geometric sequences and several ways, including recursive functions. Int: Aryabhatta is sometimes considered the series; sum of finite and infinite geometric Link infinite geometric series with limits of "father of algebra". Compare with series. convergence in 6.1. al-Khawarizmi. Sigma notation. Int: The use of several alphabets in Applications. Examples include compound interest and mathematical notation (eg first term and population growth. common difference of an arithmetic sequence). TOK: Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition? TOK: Mathematics and the world. Some mathematical constants ( π , e, φ , Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge? TOK: Mathematics and the knower. How is mathematical intuition used as a basis for formal proof? (Gauss' method for adding up integers from 1 to 100.) (continued)17 18 Content Further guidance Links (see notes above) Aim 8: Short-term loans at high interest rates. How can knowledge of mathematics result in Syllabus content individuals being exploited or protected from extortion? Appl: Physics 7.2, 13.2 (radioactive decay and nuclear physics). 1.2 Exponents and logarithms. Exponents and logarithms are further Appl: Chemistry 18.1, 18.2 (calculation of pH developed in 2.4. and buffer solutions). Laws of exponents; laws of logarithms. TOK: The nature of mathematics and science. Change of base. Were logarithms an invention or discovery? (This topic is an opportunity for teachers and students to reflect on "the nature of mathematics".) 1.3 Counting principles, including permutations n TOK: The nature of mathematics. The and combinations. The ability to find   and n Pr using both the unforeseen links between Pascal's triangle, r counting methods and the coefficients of formula and technology is expected. Link to polynomials. Is there an underlying truth that 5.4. can be found linking these? The binomial theorem: Link to 5.6, binomial distribution. Int: The properties of Pascal's triangle were known in a number of different cultures long expansion of (a + b) n , n ∈  . before Pascal (eg the Chinese mathematician Not required: Yang Hui). Permutations where some objects are identical. Aim 8: How many different tickets are Circular arrangements. possible in a lottery? What does this tell us about the ethics of selling lottery tickets to Proof of binomial theorem. those who do not understand the implicationsMathematics HL guide of these large numbers? Content Further guidance Links 1.4 Proof by mathematical induction. Links to a wide variety of topics, for example, TOK: Nature of mathematics and science. complex numbers, differentiation, sums of What are the different meanings of induction in series and divisibility. mathematics and science? TOK: Knowledge claims in mathematics. DoMathematics HL guide proofs provide us with completely certain knowledge? TOK: Knowledge communities. Who judges the validity of a proof? 1.5 Complex numbers: the number i= −1 ; the When solving problems, students may need to Appl: Concepts in electrical engineering. terms real part, imaginary part, conjugate, use technology. Impedance as a combination of resistance and modulus and argument. reactance; also apparent power as a combination of real and reactive powers. These Cartesian form z= a + ib . combinations take the form z= a + ib . Sums, products and quotients of complex TOK: Mathematics and the knower. Do the numbers. words imaginary and complex make the concepts more difficult than if they had different names? TOK: The nature of mathematics. Has "i" been invented or was it discovered? TOK: Mathematics and the world. Why does "i" appear in so many fundamental laws of physics? Syllabus content19 20 Content Further guidance Links 1.6 Modulus–argument (polar) form r eiθ is also known as Euler's form. Appl: Concepts in electrical engineering. z =r (cos θ + i sin θ ) =r cisθ =r eiθ . Phase angle/shift, power factor and apparent The ability to convert between forms is power as a complex quantity in polar form. Syllabus content expected. TOK: The nature of mathematics. Was the The complex plane. The complex plane is also known as the complex plane already there before it was used Argand diagram. to represent complex numbers geometrically? TOK: Mathematics and the knower. Why might it be said that ei π + 1 = is beautiful? 0 1.7 Powers of complex numbers: de Moivre's Proof by mathematical induction for n ∈  + . TOK: Reason and mathematics. What is theorem. mathematical reasoning and what role does proof play in this form of reasoning? Are there nth roots of a complex number. examples of proof that are not mathematical? 1.8 Conjugate roots of polynomial equations with Link to 2.5 and 2.7. real coefficients. 1.9 Solutions of systems of linear equations (a These systems should be solved using both TOK: Mathematics, sense, perception and maximum of three equations in three algebraic and technological methods, eg row reason. If we can find solutions in higher unknowns), including cases where there is a reduction. dimensions, can we reason that these spaces unique solution, an infinity of solutions or no exist beyond our sense perception? Systems that have solution(s) may be referred solution. to as consistent. When a system has an infinity of solutions, a general solution may be required. Link to vectors in 4.7.Mathematics HL guide Topic 2—Core: Functions and equations 22 hours The aims of this topic are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of technology in both the development and the application of this topic. Content Further guidance LinksMathematics HL guide 2.1 Concept of function f : x  f ( x) : domain, Int: The notation for functions was developed range; image (value). by a number of different mathematicians in the 17th and 18th centuries. How did the notation Odd and even functions. we use today become internationally accepted? Composite functions f  g . ( f  g )( x) = f ( g ( x)) . Link with 6.2. TOK: The nature of mathematics. Is mathematics simply the manipulation of Identity function. symbols under a set of formal rules? One-to-one and many-to-one functions. Link with 3.4. Inverse function f −1 , including domain Link with 6.2. restriction. Self-inverse functions. Syllabus content21 22 Content Further guidance Links 2.2 The graph of a function; its equation y = f ( x) . TOK: Mathematics and knowledge claims. Does studying the graph of a function contain Investigation of key features of graphs, such as Use of technology to graph a variety of the same level of mathematical rigour as Syllabus content maximum and minimum values, intercepts, functions. studying the function algebraically horizontal and vertical asymptotes and symmetry, (analytically)? and consideration of domain and range. Appl: Sketching and interpreting graphs; The graphs of the functions y = f ( x) and Geography SL/HL (geographic skills); Chemistry 11.3.1. y = f(x). Int: Bourbaki group analytical approach versus 1 Mandlebrot visual approach. The graph of y = given the graph of f ( x) y = f ( x) . 2.3 Transformations of graphs: translations; Link to 3.4. Students are expected to be aware Appl: Economics SL/HL 1.1 (shift in demand stretches; reflections in the axes. of the effect of transformations on both the and supply curves). algebraic expression and the graph of a The graph of the inverse function as a function. reflection in y = x . 2.4 ax + b The reciprocal function is a particular case. The rational function x  , and its cx + d Graphs should include both asymptotes and graph. any intercepts with axes. The function x  a x , a > 0 , and its graph. Exponential and logarithmic functions as Appl: Geography SL/HL (geographic skills); inverses of each other. Physics SL/HL 7.2 (radioactive decay); The function x  log a x , x > 0 , and its graph. Chemistry SL/HL 16.3 (activation energy); Link to 6.2 and the significance of e. Economics SL/HL 3.2 (exchange rates). Application of concepts in 2.1, 2.2 and 2.3.Mathematics HL guide Content Further guidance Links 2.5 Polynomial functions and their graphs. The graphical significance of repeated factors. The factor and remainder theorems. The relationship between the degree of a polynomial function and the possible numbers The fundamental theorem of algebra. of x-intercepts.Mathematics HL guide 2.6 Solving quadratic equations using the quadratic May be referred to as roots of equations or Appl: Chemistry 17.2 (equilibrium law). formula. zeros of functions. Appl: Physics 2.1 (kinematics). 2 = Use of the discriminant ∆ b − 4ac to Appl: Physics 4.2 (energy changes in simple determine the nature of the roots. harmonic motion). Solving polynomial equations both graphically Link the solution of polynomial equations to Appl: Physics (HL only) 9.1 (projectile and algebraically. conjugate roots in 1.8. motion). Sum and product of the roots of polynomial n r Aim 8: The phrase "exponential growth" is equations. For the polynomial equation ∑a x r =0, used popularly to describe a number of r =0 phenomena. Is this a misleading use of a − an −1 mathematical term? the sum is , an (−1) n a0 the product is . an Solution of a x = b using logarithms. Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Syllabus content23 24 Content Further guidance Links 2.7 Solutions of g ( x) ≥ f ( x) . Graphical or algebraic methods, for simple Syllabus content polynomials up to degree 3. Use of technology for these and other functions.Mathematics HL guide Topic 3—Core: Circular functions and trigonometry 22 hours The aims of this topic are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry. On examination papers, radian measure should be assumed unless otherwise indicated, for example, by x  sin x° . Content Further guidance LinksMathematics HL guide 3.1 The circle: radian measure of angles. Radian measure may be expressed as multiples Int: The origin of degrees in the mathematics of π, or decimals. Link with 6.2. of Mesopotamia and why we use minutes and Length of an arc; area of a sector. seconds for time. 3.2 Definition of cos θ , sin θ and tan θ in terms TOK: Mathematics and the knower. Why do of the unit circle. we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the Exact values of sin, cos and tan of implications of using these two measures on π π π π the shape of sinusoidal graphs.) 0, , , , and their multiples. 6 4 3 2 TOK: Mathematics and knowledge claims. If Definition of the reciprocal trigonometric trigonometry is based on right triangles, how ratios secθ , cscθ and cotθ . can we sensibly consider trigonometric ratios of angles greater than a right angle? Pythagorean identities: cos 2 θ + sin 2 θ = 1; 2 2 2 2 Int: The origin of the word "sine". 1 + tan θ =; 1 + cot θ =. sec θ csc θ Appl: Physics SL/HL 2.2 (forces and 3.3 Compound angle identities. Derivation of double angle identities from dynamics). compound angle identities. Appl: Triangulation used in the Global Double angle identities. Finding possible values of trigonometric ratios Positioning System (GPS). Not required: without finding θ, for example, finding sin 2θ Int: Why did Pythagoras link the study of Proof of compound angle identities. given sin θ . music and mathematics? Appl: Concepts in electrical engineering. Generation of sinusoidal voltage. (continued) Syllabus content25 26 Content Further guidance Links 3.4 Composite functions of the form (see notes above) f= a sin(b( x + c)) + d . ( x) TOK: Mathematics and the world. Music can be expressed using mathematics. Does this Syllabus content Applications. mean that music is mathematical, that mathematics is musical or that both are 3.5 The inverse functions x  arcsin x , reflections of a common "truth"? x  arccos x , x  arctan x ; their domains and Appl: Physics SL/HL 4.1 (kinematics of ranges; their graphs. simple harmonic motion). 3.6 Algebraic and graphical methods of solving TOK: Mathematics and knowledge claims. trigonometric equations in a finite interval, How can there be an infinite number of including the use of trigonometric identities discrete solutions to an equation? and factorization. Not required: The general solution of trigonometric equations. 3.7 The cosine rule TOK: Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or The sine rule including the ambiguous case. more than 180°, what does this tell us about the 1 "fact" of the angle sum of a triangle and about Area of a triangle as ab sin C . the nature of mathematical knowledge? 2 Applications. Examples include navigation, problems in two Appl: Physics SL/HL 1.3 (vectors and scalars); and three dimensions, including angles of Physics SL/HL 2.2 (forces and dynamics). elevation and depression. Int: The use of triangulation to find the curvature of the Earth in order to settle a dispute between England and France overMathematics HL guide Newton's gravity. Topic 4—Core: Vectors 24 hours The aim of this topic is to introduce the use of vectors in two and three dimensions, and to facilitate solving problems involving points, lines and planes. Content Further guidance Links 4.1 Concept of a vector. Aim 8: Vectors are used to solve manyMathematics HL guide problems in position location. This can be used Representation of vectors using directed line to save a lost sailor or destroy a building with a segments. laser-guided bomb. Unit vectors; base vectors i, j, k. Components of a vector: Appl: Physics SL/HL 1.3 (vectors and scalars); Physics SL/HL 2.2 (forces and dynamics).  v1    TOK: Mathematics and knowledge claims. v = v2  =v1i + v2 j + v3 k . v  You can perform some proofs using different  3 mathematical concepts. What does this tell us about mathematical knowledge? Algebraic and geometric approaches to the Proofs of geometrical properties using vectors. following: • the sum and difference of two vectors; • the zero vector 0 , the vector −v ; • multiplication by a scalar, kv ; • magnitude of a vector, v ; → • position vectors OA = a . → Distance between points A and B is the AB= b − a → magnitude of AB . Syllabus content27 30 Topic 5—Core: Statistics and probability 36 hours The aim of this topic is to introduce basic concepts. It may be considered as three parts: manipulation and presentation of statistical data (5.1), the laws of probability (5.2–5.4), and random variables and their probability distributions (5.5–5.7). It is expected that most of the calculations required will be done on a GDC. The emphasis is on understanding and interpreting the results obtained. Statistical tables will no longer be allowed in examinations. Syllabus content Content Further guidance Links 5.1 Concepts of population, sample, random For examination purposes, in papers 1 and 2 TOK: The nature of mathematics. Why have sample and frequency distribution of discrete data will be treated as the population. mathematics and statistics sometimes been and continuous data. treated as separate subjects? In examinations the following formulae should Grouped data: mid-interval values, interval be used: TOK: The nature of knowing. Is there a width, upper and lower interval boundaries. k difference between information and data? Mean, variance, standard deviation. ∑fx i i Aim 8: Does the use of statistics lead to an i =1 µ= , overemphasis on attributes that can easily be Not required: n measured over those that cannot? Estimation of mean and variance of a k k 2 population from a sample. ∑ f (x i − µ )2 i ∑fx i i Appl: Psychology SL/HL (descriptive 2 i =1 i =1 2 statistics); Geography SL/HL (geographic σ = = −µ . n n skills); Biology SL/HL 1.1.2 (statistical analysis). Appl: Methods of collecting data in real life (census versus sampling). Appl: Misleading statistics in media reports.Mathematics HL guide Content Further guidance Links 5.2 Concepts of trial, outcome, equally likely Aim 8: Why has it been argued that theories outcomes, sample space (U) and event. based on the calculable probabilities found in n( A) casinos are pernicious when applied to The probability of an event A as P( A) = . everyday life (eg economics)? n(U )Mathematics HL guide Int: The development of the mathematical The complementary events A and A′ (not A). theory of probability in 17th century France. Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems. 5.3 Combined events; the formula for P( A ∪ B ) . Mutually exclusive events. 5.4 Conditional probability; the definition Appl: Use of probability methods in medical P( A ∩ B ) studies to assess risk factors for certain P( A \| B) = . diseases. P( B ) TOK: Mathematics and knowledge claims. Is Independent events; the definition Use of P( A ∩ B ) =A)P( B ) to show P( independence as defined in probabilistic terms = = P ( A \| B ) P ( A ) P ( A \| B′ ) . independence. the same as that found in normal experience? Use of Bayes' theorem for a maximum of three events. Syllabus content31 32 Content Further guidance Links 5.5 Concept of discrete and continuous random TOK: Mathematics and the knower. To what variables and their probability distributions. extent can we trust samples of data? Syllabus content Definition and use of probability density functions. Expected value (mean), mode, median, For a continuous random variable, a value at variance and standard deviation. which the probability density function has a maximum value is called a mode. Applications. Examples include games of chance. Appl: Expected gain to insurance companies. 5.6 Binomial distribution, its mean and variance. Link to binomial theorem in 1.3. TOK: Mathematics and the real world. Is the binomial distribution ever a useful model for Poisson distribution, its mean and variance. Conditions under which random variables have an actual real-world situation? these distributions. Not required: Formal proof of means and variances. 5.7 Normal distribution. Probabilities and values of the variable must be Appl: Chemistry SL/HL 6.2 (collision theory); found using technology. Psychology HL (descriptive statistics); Biology SL/HL 1.1.3 (statistical analysis). The standardized value (z) gives the number of standard deviations from the mean. Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and Properties of the normal distribution. Link to 2.3. conclusions? Standardization of normal variables. TOK: Mathematics and knowledge claims. To what extent can we trust mathematical models such as the normal distribution? Int: De Moivre's derivation of the normal distribution and Quetelet's use of it to describeMathematics HL guide l'homme moyen. Topic 6—Core: Calculus 48 hours The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their application. Content Further guidance Links 6.1 Informal ideas of limit, continuity and sin θ TOK: The nature of mathematics. Does theMathematics HL guide convergence. Include result lim =1. fact that Leibniz and Newton came across the θ →0 θ calculus at similar times support the argument Definition of derivative from first principles Link to 1.1. that mathematics exists prior to its discovery? f ( x + h) − f ( x ) Use of this definition for polynomials only. f ′( x) = lim . Int: How the Greeks' distrust of zero meant h →0 h Link to binomial theorem in 1.3. that Archimedes' work did not lead to calculus. The derivative interpreted as a gradient Int: Investigate attempts by Indian dy function and as a rate of change. Both forms of notation, and f ′ ( x ) , for the mathematicians (500–1000 CE) to explain dx Finding equations of tangents and normals. division by zero. first derivative. Identifying increasing and decreasing TOK: Mathematics and the knower. What functions. does the dispute between Newton and Leibniz tell us about human emotion and mathematical The second derivative. Use of both algebra and technology. discovery? Higher derivatives. Appl: Economics HL 1.5 (theory of the firm); Chemistry SL/HL 11.3.4 (graphical d2 y techniques); Physics SL/HL 2.1 (kinematics). Both forms of notation, and f ′′( x) , for dx 2 the second derivative. dn y Familiarity with the notation and dx n f ( n ) ( x) . Link with induction in 1.4. Syllabus content33 34 Content Further guidance Links 6.2 Derivatives of x n , sin x , cos x , tan x , e x and Appl: Physics HL 2.4 (uniform circular motion); ln x . Physics 12.1 (induced electromotive force (emf)). Syllabus content Differentiation of sums and multiples of TOK: Mathematics and knowledge claims. functions. Euler was able to make important advances in mathematical analysis before calculus had been The product and quotient rules. put on a solid theoretical foundation by Cauchy The chain rule for composite functions. and others. However, some work was not possible until after Cauchy's work. What does Related rates of change. this tell us about the importance of proof and Implicit differentiation. the nature of mathematics? Derivatives of sec x , csc x , cot x , a x , log a x , TOK: Mathematics and the real world. The seemingly abstract concept of calculus allows us arcsin x , arccos x and arctan x . to create mathematical models that permit human feats, such as getting a man on the Moon. What does this tell us about the links between mathematical models and physical reality? 6.3 Local maximum and minimum values. Testing for the maximum or minimum using the change of sign of the first derivative and Optimization problems. using the sign of the second derivative. Points of inflexion with zero and non-zero gradients. Use of the terms "concave up" for f ′′( x) > 0 , "concave down" for f ′′( x) < 0 . Graphical behaviour of functions, including the relationship between the graphs of At a point of inflexion, f ′′( x) = 0 and changes f , f ′ and f ′′ . sign (concavity change). Not required: Points of inflexion, where f ′′( x) is notMathematics HL guide defined, for example, y = x1 3 at (0,0) . Content Further guidance Links 6.4 Indefinite integration as anti-differentiation. Indefinite integral interpreted as a family of curves. Indefinite integral of x n , sin x , cos x and e x . 1 = ln x + c . dx Other indefinite integrals using the results from ∫ x 6.2. 5Mathematics HL guide 1 Examples include ∫ ( 2 x − 1) dx , ∫ dx The composites of any of these with a linear 3x + 4 function. 1 and ∫ 2 dx . x + 2x + 5 6.5 Anti-differentiation with a boundary condition to determine the constant of integration. Definite integrals. The value of some definite integrals can only be found using technology. Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves. Volumes of revolution about the x-axis or y-axis. Appl: Industrial design. Syllabus content35 Topic 7—Option: Statistics and probability 48 hours The aims of this option are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding; and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option, and that the minimum requirement of a GDC will be to find probability distribution function (pdf), cumulative distribution function (cdf), inverse cumulative distribution function, p-values and test statistics, including calculations for the following distributions: binomial, Poisson, normal and t. Students are expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their written answers. Calculator-specific orMathematics HL guide brand-specific language should not be used within these explanations. Content Further guidance Links 7.1 Cumulative distribution functions for both discrete and continuous distributions. Geometric distribution. Negative binomial distribution. = E(t X ) G (t ) = P( X ∑= x)t x . Int: Also known as Pascal's distribution. x Probability generating functions for discrete random variables. Using probability generating functions to find Aim 8: Statistical compression of data files. mean, variance and the distribution of the sum of n independent random variables. 7.2 Linear transformation of a single random variable. E(aX += aE( X ) + b , b) Mean of linear combinations of n random Var(aX + b) =Var( X ) . a2 variables. Variance of linear combinations of n independent random variables. Expectation of the product of independent E( XY ) = E( X )E(Y ) . random variables. Syllabus content37 38 Content Further guidance Links 7.3 Unbiased estimators and estimates. T is an unbiased estimator for the parameter TOK: Mathematics and the world. In the θ if E(T ) = θ . absence of knowing the value of a parameter, Comparison of unbiased estimators based on will an unbiased estimator always be better Syllabus content variances. T1 is a more efficient estimator than T2 if than a biased one? Var(T1 ) < Var(T2 ) . n X as an unbiased estimator for µ . Xi X =∑ . i =1 n S 2 as an unbiased estimator for σ 2 . 2 n 2 (X i −X) S =∑ . i =1 n −1 7.4 A linear combination of independent normal Aim 8/TOK: Mathematics and the world. random variables is normally distributed. In "Without the central limit theorem, there could particular, be no statistics of any value within the human  σ2  sciences." X ~ N( µ ,σ 2 ) ⇒ X ~ N  µ ,  .  n  TOK: Nature of mathematics. The central limit theorem can be proved mathematically The central limit theorem. (formalism), but its truth can be confirmed by its applications (empiricism).Mathematics HL guide Content Further guidance Links 7.5 Confidence intervals for the mean of a normal Use of the normal distribution when σ is TOK: Mathematics and the world. Claiming population. known and use of the t-distribution when σ is brand A is "better" on average than brand B unknown, regardless of sample size. The case can mean very little if there is a large overlap of matched pairs is to be treated as an example between the confidence intervals of the two of a single sample technique. means.Mathematics HL guide Appl: Geography. 7.6 Null and alternative hypotheses, H 0 and H1 . Use of the normal distribution when σ is TOK: Mathematics and the world. In practical known and use of the t-distribution when σ is terms, is saying that a result is significant the Significance level. unknown, regardless of sample size. The case same as saying that it is true? Critical regions, critical values, p-values, one- of matched pairs is to be treated as an example TOK: Mathematics and the world. Does the tailed and two-tailed tests. of a single sample technique. ability to test only certain parameters in a Type I and II errors, including calculations of population affect the way knowledge claims in their probabilities. the human sciences are valued? Testing hypotheses for the mean of a normal Appl: When is it more important not to make a population. Type I error and when is it more important not to make a Type II error? Syllabus content39 40 Content Further guidance Links 7.7 Introduction to bivariate distributions. Informal discussion of commonly occurring Appl: Geographic skills. situations, eg marks in pure mathematics and Aim 8: The correlation between smoking and statistics exams taken by a class of students, Syllabus content lung cancer was "discovered" using salary and age of teachers in a certain school. mathematics. Science had to justify the cause. The need for a measure of association between the variables and the possibility of predicting the value of one of the variables given the value of the other variable. Covariance and (population) product moment Cov( X , Y ) = E[( X − µ x )(Y − µ y )] Appl: Using technology to fit a range of curves correlation coefficient ρ. to a set of data. = E( XY ) − µ x µ y , = = where µ x E( X ), µ y E(Y ) . Cov( X , Y ) ρ= . Var( X )Var(Y ) Proof that ρ = 0 in the case of independence The use of ρ as a measure of association TOK: Mathematics and the world. Given that a and ±1 in the case of a linear relationship between X and Y, with values near 0 indicating set of data may be approximately fitted by a between X and Y. a weak association and values near +1 or near range of curves, where would we seek for –1 indicating a strong association. knowledge of which equation is the "true" model? Definition of the (sample) product moment n Aim 8: The physicist Frank Oppenheimer correlation coefficient R in terms of n paired ∑(X i − X )(Yi − Y ) wrote: "Prediction is dependent only on the i =1 observations on X and Y. Its application to the R= assumption that observed patterns will be n n estimation of ρ. ∑(X i − X ) 2 ∑ (Yi − Y ) 2 repeated." This is the danger of extrapolation. i =1 i =1 There are many examples of its failure in the n past, eg share prices, the spread of disease, i i ∑ X Y − nXY climate change. i =1 = .Mathematics HL guide  n 2 2  2 2  ∑ X i − n X  ∑ Yi − nY  (continued)  i =1   42 Topic 8—Option: Sets, relations and groups 48 hours The aims of this option are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra. Syllabus content Content Further guidance Links 8.1 Finite and infinite sets. Subsets. TOK: Cantor theory of transfinite numbers, Russell's paradox, Godel's incompleteness Operations on sets: union; intersection; theorems. complement; set difference; symmetric difference. De Morgan's laws: distributive, associative and Illustration of these laws using Venn diagrams. Appl: Logic, Boolean algebra, computer commutative laws (for union and intersection). circuits. Students may be asked to prove that two sets are the same by establishing that A ⊆ B and B ⊆ A . 8.2 Ordered pairs: the Cartesian product of two sets. Relations: equivalence relations; equivalence An equivalence relation on a set forms a Appl, Int: Scottish clans. classes. partition of the set. 8.3 Functions: injections; surjections; bijections. The term codomain. Composition of functions and inverse Knowledge that the function composition is not functions. a commutative operation and that if f is a bijection from set A to set B then f −1 exists and is a bijection from set B to set A.Mathematics HL guide Content Further guidance Links 8.4 Binary operations. A binary operation ∗ on a non-empty set S is a rule for combining any two elements a, b ∈ S to give a unique element c. That is, in this definition, a binary operation on a set is not necessarily closed.Mathematics HL guide Operation tables (Cayley tables). 8.5 Binary operations: associative, distributive and The arithmetic operations on  and . TOK: Which are more fundamental, the commutative properties. general models or the familiar examples? Examples of distributivity could include the fact that, on  , multiplication is distributive over addition but addition is not distributive over multiplication. 8.6 The identity element e. Both the right-identity a ∗ e = and left- a identity e ∗ a = must hold if e is an identity a The inverse a −1 of an element a. element. Proof that left-cancellation and right- cancellation by an element a hold, provided Both a ∗ a −1 = a −1 ∗ a = must hold. e and e that a has an inverse. Proofs of the uniqueness of the identity and inverse elements. Syllabus content43 44 Content Further guidance Links 8.7 The definition of a group {G , ∗} . For the set G under a given operation ∗ : Appl: Existence of formula for roots of polynomials. The operation table of a group is a Latin • G is closed under ∗ ; Syllabus content square, but the converse is false. Appl: Galois theory for the impossibility of • ∗ is associative; such formulae for polynomials of degree 5 or • G contains an identity element; higher. • each element in G has an inverse in G. Abelian groups. a ∗ b = b ∗ a , for all a, b ∈ G . 8.8 Examples of groups: Appl: Rubik's cube, time measures, crystal structure, symmetries of molecules, strut and • , ,  and  under addition; cable constructions, Physics H2.2 (special relativity), the 8–fold way, supersymmetry. • integers under addition modulo n; • non-zero integers under multiplication, modulo p, where p is prime; symmetries of plane figures, including The composition T2  T1 denotes T1 followed equilateral triangles and rectangles; by T2 . invertible functions under composition of functions. 8.9 The order of a group. The order of a group element. Cyclic groups. Appl: Music circle of fifths, prime numbers. Generators. Proof that all cyclic groups are Abelian.Mathematics HL guide Content Further guidance Links 8.10 Permutations under composition of On examination papers: the form Appl: Cryptography, campanology. permutations. 1 2 3 p=  or in cycle notation (132) will Cycle notation for permutations. 3 1 2 Result that every permutation can be written as be used to represent the permutation 1 → 3 ,Mathematics HL guide a composition of disjoint cycles. 2 → 1 , 3 → 2. The order of a combination of cycles. 8.11 Subgroups, proper subgroups. A proper subgroup is neither the group itself nor the subgroup containing only the identity element. Use and proof of subgroup tests. Suppose that {G , ∗} is a group and H is a non-empty subset of G. Then {H , ∗} is a subgroup of {G , ∗} if a ∗ b −1 ∈ H whenever a, b ∈ H . Suppose that {G , ∗} is a finite group and H is a non-empty subset of G. Then {H , ∗} is a subgroup of {G , ∗} if H is closed under ∗ . Definition and examples of left and right cosets of a subgroup of a group. Lagrange's theorem. Appl: Prime factorization, symmetry breaking. Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange's theorem.) Syllabus content45 46 Content Further guidance Links 8.12 Definition of a group homomorphism. Infinite groups as well as finite groups. Let {G ,} and {H , } be groups, then the Syllabus content function f : G → H is a homomorphism if f (a b) = f (a )  f (b) for all a, b ∈ G . Definition of the kernel of a homomorphism. If f : G → H is a group homomorphism, then Proof that the kernel and range of a Ker( f ) is the set of a ∈ G such that homomorphism are subgroups. f (a ) = eH . Proof of homomorphism properties for Identity: let eG and eH be the identity elements identities and inverses. of (G, ∗) and ( H , ) , respectively, then f (eG ) = eH . −1 Inverse: f (a −1 ) = ( f (a ) ) for all a ∈ G . Isomorphism of groups. Infinite groups as well as finite groups. The homomorphism f : G → H is an isomorphism if f is bijective. The order of an element is unchanged by an isomorphism.Mathematics HL guide Topic 9—Option: Calculus 48 hours The aims of this option are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations. Content Further guidance Links 9.1 Infinite sequences of real numbers and their Informal treatment of limit of sum, difference, TOK: Zeno's paradox, impact of infiniteMathematics HL guide convergence or divergence. product, quotient; squeeze theorem. sequences and limits on our understanding of the physical world. Divergent is taken to mean not convergent. 9.2 Convergence of infinite series. The sum of a series is the limit of the sequence 1 TOK: Euler's idea that 1 − 1 + 1 − 1 +  = . 2 of its partial sums. Was it a mistake or just an alternative view? Tests for convergence: comparison test; limit comparison test; ratio test; integral test. Students should be aware that if lim xn = 0 x →∞ then the series is not necessarily convergent, but if lim xn ≠ 0 , the series diverges. x →∞ 1 1 The p-series, ∑n p . ∑n p is convergent for p > 1 and divergent otherwise. When p = 1 , this is the harmonic series. Series that converge absolutely. Conditions for convergence. Series that converge conditionally. Alternating series. The absolute value of the truncation error is less than the next term in the series. Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test. Syllabus content47 Topic 10—Option: Discrete mathematics 48 hours The aim of this option is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications. Content Further guidance Links 10.1 Strong induction. For example, proofs of the fundamental TOK: Mathematics and knowledge claims.Mathematics HL guide theorem of arithmetic and the fact that a tree The difference between proof and conjecture, Pigeon-hole principle. with n vertices has n – 1 edges. eg Goldbach's conjecture. Can a mathematical statement be true before it is proven? TOK: Proof by contradiction. 10.2 na a \| b ⇒ b = for some n ∈  . The theorem a \| b and a \| c ⇒ a \| (bx ± cy ) The division algorithm = bq + r , 0 ≤ r < b . a where x, y ∈  . Division and Euclidean algorithms. The Euclidean algorithm for determining the Int: Euclidean algorithm contained in Euclid's greatest common divisor of two integers. Elements, written in Alexandria about The greatest common divisor, gcd(a, b) , and 300 BCE. the least common multiple, lcm(a, b) , of integers a and b. Aim 8: Use of prime numbers in cryptography. The possible impact of the discovery of Prime numbers; relatively prime numbers and powerful factorization techniques on internet the fundamental theorem of arithmetic. and bank security. 10.3 Linear Diophantine equations ax + by = c. General solutions required and solutions Int: Described in Diophantus' Arithmetica subject to constraints. For example, all written in Alexandria in the 3rd century CE. solutions must be positive. When studying Arithmetica, a French mathematician, Pierre de Fermat (1601–1665) wrote in the margin that he had discovered a simple proof regarding higher-order Diophantine equations—Fermat's last theorem. Syllabus content51 52 Content Further guidance Links 10.4 Modular arithmetic. The solution of linear congruences. Syllabus content Solution of simultaneous linear congruences Int: Discussed by Chinese mathematician Sun (Chinese remainder theorem). Tzu in the 3rd century CE. 10.5 Representation of integers in different bases. On examination papers, questions that go Int: Babylonians developed a base 60 number beyond base 16 will not be set. system and the Mayans a base 20 number system. 10.6 Fermat's little theorem. a p = a (mod p ) , where p is prime. TOK: Nature of mathematics. An interest may be pursued for centuries before becoming "useful".Mathematics HL guide Content Further guidance Links 10.7 Graphs, vertices, edges, faces. Adjacent Two vertices are adjacent if they are joined by Aim 8: Symbolic maps, eg Metro and vertices, adjacent edges. an edge. Two edges are adjacent if they have a Underground maps, structural formulae in common vertex. chemistry, electrical circuits. Degree of a vertex, degree sequence. TOK: Mathematics and knowledge claims. Handshaking lemma.Mathematics HL guide Proof of the four-colour theorem. If a theorem is proved by computer, how can we claim to know that it is true? Simple graphs; connected graphs; complete It should be stressed that a graph should not be Aim 8: Importance of planar graphs in graphs; bipartite graphs; planar graphs; trees; assumed to be simple unless specifically stated. constructing circuit boards. weighted graphs, including tabular The term adjacency table may be used. representation. Subgraphs; complements of graphs. Euler's relation: v − e + f = ; theorems for 2 If the graph is simple and planar and v ≥ 3 , TOK: Mathematics and knowledge claims. planar graphs including e ≤ 3v − 6 , e ≤ 2v − 4 , then e ≤ 3v − 6 . Applications of the Euler characteristic leading to the results that κ 5 and κ 3,3 are not (v − e + f ) to higher dimensions. Its use in If the graph is simple, planar, has no cycles of planar. length 3 and v ≥ 3 , then e ≤ 2v − 4 . understanding properties of shapes that cannot be visualized. 10.8 Walks, trails, paths, circuits, cycles. Eulerian trails and circuits. A connected graph contains an Eulerian circuit Int: The "Bridges of Königsberg" problem. if and only if every vertex of the graph is of even degree. Hamiltonian paths and cycles. Simple treatment only. 10.9 Graph algorithms: Kruskal's; Dijkstra's. Syllabus content53 54 Content Further guidance Links 10.10 Chinese postman problem. To determine the shortest route around a Int: Problem posed by the Chinese weighted graph going along each edge at least mathematician Kwan Mei-Ko in 1962. Not required: once. Syllabus content Graphs with more than four vertices of odd degree. Travelling salesman problem. To determine the Hamiltonian cycle of least TOK: Mathematics and knowledge claims. weight in a weighted complete graph. How long would it take a computer to test all Nearest-neighbour algorithm for determining Hamiltonian cycles in a complete, weighted an upper bound. graph with just 30 vertices? Deleted vertex algorithm for determining a lower bound. 10.11 Recurrence relations. Initial conditions, TOK: Mathematics and the world. The recursive definition of a sequence. connections of sequences such as the Fibonacci sequence with art and biology. Solution of first- and second-degree linear Includes the cases where auxiliary equation has homogeneous recurrence relations with equal roots or complex roots. constant coefficients. The first-degree linear recurrence relation = aun −1 + b . un Modelling with recurrence relations. Solving problems such as compound interest, debt repayment and counting problems.Mathematics HL guide SyllabusGlossary of terminology: Discrete mathematicsIntroductionTeachers and students should be aware that many different terminologies exist in graph theory, and thatdifferent textbooks may employ different combinations of these. Examples of these are: vertex/node/junction/point; edge/route/arc; degree/order of a vertex; multiple edges/parallel edges; loop/self-loop.In IB examination questions, the terminology used will be as it appears in the syllabus. For clarity, these termsare defined below.TerminologyBipartite graph A graph whose vertices can be divided into two sets such that no two vertices in the same set are adjacent.Circuit A walk that begins and ends at the same vertex, and has no repeated edges.Complement of a A graph with the same vertices as G but which has an edge between any twograph G vertices if and only if G does not.Complete bipartite A bipartite graph in which every vertex in one set is joined to every vertex in thegraph other set.Complete graph A simple graph in which each pair of vertices is joined by an edge.Connected graph A graph in which each pair of vertices is joined by a path.Cycle A walk that begins and ends at the same vertex, and has no other repeated vertices.Degree of a vertex The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points.Disconnected graph A graph that has at least one pair of vertices not joined by a path.Eulerian circuit A circuit that contains every edge of a graph.Eulerian trail A trail that contains every edge of a graph.Graph Consists of a set of vertices and a set of edges.Graph isomorphism A one-to-one correspondence between vertices of G and H such that a pair ofbetween two simple vertices in G is adjacent if and only if the corresponding pair in H is adjacent.graphs G and HHamiltonian cycle A cycle that contains all the vertices of the graph.Hamiltonian path A path that contains all the vertices of the graph.Loop An edge joining a vertex to itself. Mathematics HL guide 55 Glossary of terminology: Discrete mathematicsMinimum spanning A spanning tree of a weighted graph that has the minimum total weight.treeMultiple edges Occur if more than one edge joins the same pair of vertices.Path A walk with no repeated vertices.Planar graph A graph that can be drawn in the plane without any edge crossing another.Simple graph A graph without loops or multiple edges.Spanning tree of a A subgraph that is a tree, containing every vertex of the graph.graphSubgraph A graph within a graph.Trail A walk in which no edge appears more than once.Tree A connected graph that contains no cycles.Walk A sequence of linked edges.Weighted graph A graph in which each edge is allocated a number or weight.Weighted tree A tree in which each edge is allocated a number or weight.56 Mathematics HL guide AssessmentAssessment in the Diploma ProgrammeGeneralAssessment is an integral part of teaching and learning. The most important aims of assessment in the DiplomaProgramme are that it should support curricular goals and encourage appropriate student learning. Bothexternal and internal assessment are used in the Diploma Programme. IB examiners mark work producedfor external assessment, while work produced for internal assessment is marked by teachers and externallymoderated by the IB.There are two types of assessment identified by the IB.• Formative assessment informs both teaching and learning. It is concerned with providing accurate and helpful feedback to students and teachers on the kind of learning taking place and the nature of students' strengths and weaknesses in order to help develop students' understanding and capabilities. Formative assessment can also help to improve teaching quality, as it can provide information to monitor progress towards meeting the course aims and objectives.• Summative assessment gives an overview of previous learning and is concerned with measuring student achievement.The Diploma Programme primarily focuses on summative assessment designed to record student achievementat or towards the end of the course of study. However, many of the assessment instruments can also beused formatively during the course of teaching and learning, and teachers are encouraged to do this. Acomprehensive assessment plan is viewed as being integral with teaching, learning and course organization.For further information, see the IB Programme standards and practices document.The approach to assessment used by the IB is criterion-related, not norm-referenced. This approach toassessment judges students' work by their performance in relation to identified levels of attainment, and not inrelation to the work of other students. For further information on assessment within the Diploma Programme,please refer to the publication Diploma Programme assessment: Principles and practice.To support teachers in the planning, delivery and assessment of the Diploma Programme courses, a varietyof resources can be found on the OCC or purchased from the IB store ( Teacher supportmaterials, subject reports, internal assessment guidance, grade descriptors, as well as resources from otherteachers, can be found on the OCC. Specimen and past examination papers as well as markschemes can bepurchased from the IB store. Mathematics HL guide 57 Assessment in the Diploma ProgrammeMethods of assessmentThe IB uses several methods to assess work produced by students.Assessment criteriaAssessment criteria are used when the assessment task is open-ended. Each criterion concentrates on aparticular skill that students are expected to demonstrate. An assessment objective describes what studentsshould be able to do, and assessment criteria describe how well they should be able to do it. Using assessmentcriteria allows discrimination between different answers and encourages a variety of responses. Each criterioncomprises a set of hierarchically ordered level descriptors. Each level descriptor is worth one or more marks.Each criterion is applied independently using a best-fit model. The maximum marks for each criterion maydiffer according to the criterion's importance. The marks awarded for each criterion are added together to givethe total mark for the piece of work.MarkbandsMarkbands are a comprehensive statement of expected performance against which responses are judged. Theyrepresent a single holistic criterion divided into level descriptors. Each level descriptor corresponds to a rangeof marks to differentiate student performance. A best-fit approach is used to ascertain which particular mark touse from the possible range for each level descriptor.MarkschemesThis generic term is used to describe analytic markschemes that are prepared for specific examination papers.Analytic markschemes are prepared for those examination questions that expect a particular kind of responseand/or a given final answer from the students. They give detailed instructions to examiners on how to breakdown the total mark for each question for different parts of the response. A markscheme may include thecontent expected in the responses to questions or may be a series of marking notes giving guidance on how toapply criteria.58 Mathematics HL guide AssessmentExternal assessmentGeneralMarkschemes are used to assess students in all papers. The markschemes are specific to each examination.External assessment detailsPapers 1, 2 and 3These papers are externally set and externally marked. Together, they contribute 80% of the final mark forthe course. These papers are designed to allow students to demonstrate what they know and what they can do.CalculatorsPaper 1Students are not permitted access to any calculator. Questions will mainly involve analytic approaches tosolutions, rather than requiring the use of a GDC. The paper is not intended to require complicated calculations,with the potential for careless errors. However, questions will include some arithmetical manipulations whenthey are essential to the development of the question.Papers 2 and 3Students must have access to a GDC at all times. However, not all questions will necessarily require the use ofthe GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for theDiploma Programme.Mathematics HL and further mathematics HL formula bookletEach student must have access to a clean copy of the formula booklet during the examination. It is theresponsibility of the school to download a copy from IBIS or the OCC and to ensure that there are sufficientcopies available for all students.Awarding of marksMarks may be awarded for method, accuracy, answers and reasoning, including interpretation.In paper 1 and paper 2, full marks are not necessarily awarded for a correct answer with no working.Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphsor calculations). Where an answer is incorrect, some marks may be given for correct method, provided this isshown by written working. All students should therefore be advised to show their working.60 Mathematics HL guide External assessmentPaper 1Duration: 2 hoursWeighting: 30%• This paper consists of section A, short-response questions, and section B, extended-response questions.• Students are not permitted access to any calculator on this paper. 2Duration: 2 hoursWeighting: 30%• This paper consists of section A, short-response questions, and section B, extended-response questions.• A GDC is required for this paper, but not every question will necessarily require its use. Mathematics HL guide 61 External assessment 3Duration: 1 hourWeighting: 20%• This paper consists of a small number of compulsory extended-response questions based on the option chosen.• Where possible, the first part of each question will be on core material leading to the option topic. When this is not readily achievable, as, for example, with the discrete mathematics option, the level of difficulty of the earlier part of a question will be comparable to that of the core questions.Syllabus coverage• Students must answer all questions.• Knowledge of the entire content of the option studied, as well as the core material, is required for this paper.62 Mathematics HL guide External assessmentMark allocation• This paper is worth 60 marks, representing 20% of the final mark.• Questions may be unequal in terms of length and level of difficulty. Therefore, individual questions may not be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of each question.Question type• Questions require extended responses involving sustained reasoning.• Individual questions will develop a single theme or be divided into unconnected parts. Where the latter occur, the unconnected parts will be clearly labelled as such Mathematics HL guide 63 AssessmentInternal assessmentPurpose of internal assessmentInternal assessment is an integral part of the course and is compulsory for all students. It enables students todemonstrate the application of their skills and knowledge, and to pursue their personal interests, without thetime limitations and other constraints that are associated with written examinations. The internal assessmentshould, as far as possible, be woven into normal classroom teaching and not be a separate activity conductedafter a course has been taught.Internal assessment in mathematics HL is an individual exploration. This is a piece of written work thatinvolves investigating an area of mathematics. It is marked according to five assessment criteria.Guidance and authenticityThe exploration submitted for internal assessment must be the student's own work. However, it is not theintention that students should decide upon a title or topic and be left to work on the exploration without anyfurther support from the teacher. The teacher should play an important role during both the planning stage andthe period when the student is working on the exploration. It is the responsibility of the teacher to ensure thatstudents are familiar with:• the requirements of the type of work to be internally assessed• the IB academic honesty policy available on the OCC• the assessment criteria—students must understand that the work submitted for assessment must address these criteria effectively.Teachers and students must discuss the exploration. Students should be encouraged to initiate discussionswith the teacher to obtain advice and information, and students must not be penalized for seeking guidance.However, if a student could not have completed the exploration without substantial support from the teacher,this should be recorded on the appropriate form from the Handbook of procedures for the Diploma Programme.It is the responsibility of teachers to ensure that all students understand the basic meaning and significanceof concepts that relate to academic honesty, especially authenticity and intellectual property. Teachers mustensure that all student work for assessment is prepared according to the requirements and must explain clearlyto students that the exploration must be entirely their own.As part of the learning process, teachers can give advice to students on a first draft of the exploration. Thisadvice should be in terms of the way the work could be improved, but this first draft must not be heavilyannotated or edited by the teacher. The next version handed to the teacher after the first draft must be the finalone.All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must notinclude any known instances of suspected or confirmed malpractice. Each student must sign the coversheet forinternal assessment to confirm that the work is his or her authentic work and constitutes the final version ofthat work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator)for internal assessment, together with the signed coversheet, it cannot be retracted.64 Mathematics HL guide Internal assessmentAuthenticity may be checked by discussion with the student on the content of the work, and scrutiny of one ormore of the following:• the student's initial proposal• the first draft of the written work• the references cited• the style of writing compared with work known to be that of the student.The requirement for teachers and students to sign the coversheet for internal assessment applies to the work ofall students, not just the sample work that will be submitted to an examiner for the purpose of moderation. If theteacher and student sign a coversheet, but there is a comment to the effect that the work may not be authentic,the student will not be eligible for a mark in that component and no grade will be awarded. For further detailsrefer to the IB publication Academic honesty and the relevant articles in the General regulations: DiplomaProgramme.The same piece of work cannot be submitted to meet the requirements of both the internal assessment and theextended essay.Group workGroup work should not be used for explorations. Each exploration is an individual piece of work based ondifferent data collected or measurements generated.It should be made clear to students that all work connected with the exploration, including the writing of theexploration, should be their own. It is therefore helpful if teachers try to encourage in students a sense of responsibilityfor their own learning so that they accept a degree of ownership and take pride in their own work.Time allocationInternal assessment is an integral part of the mathematics HL course, contributing 20% to the final assessmentin the course. This weighting should be reflected in the time that is allocated to teaching the knowledge, skillsand understanding required to undertake the work as well as the total time allocated to carry out the work.It is expected that a total of approximately 10 teaching hours should be allocated to the work. This shouldinclude:• time for the teacher to explain to students the requirements of the exploration• class time for students to work on the exploration• time for consultation between the teacher and each student• time to review and monitor progress, and to check authenticity.Using assessment criteria for internal assessmentFor internal assessment, a number of assessment criteria have been identified. Each assessment criterion haslevel descriptors describing specific levels of achievement together with an appropriate range of marks. Thelevel descriptors concentrate on positive achievement, although for the lower levels failure to achieve may beincluded in the description. Mathematics HL guide 65 Internal assessmentTeachers must judge the internally assessed work against the criteria using the level descriptors.• The aim is to find, for each criterion, the descriptor that conveys most accurately the level attained by the student.• When assessing a student's work, teachers should read the level descriptors for each criterion, starting with level 0, until they reach a descriptor that describes a level of achievement that has not been reached. The level of achievement gained by the student is therefore the preceding one, and it is this that should be recorded.• Only whole numbers should be recorded; partial marks, that is fractions and decimals, are not acceptable.• Teachers should not think in terms of a pass or fail boundary, but should concentrate on identifying the appropriate descriptor for each assessment criterion.• The highest level descriptors do not imply faultless performance but should be achievable by a student. Teachers should not hesitate to use the extremes if they are appropriate descriptions of the work being assessed.• A student who attains a high level of achievement in relation to one criterion will not necessarily attain high levels of achievement in relation to the other criteria. Similarly, a student who attains a low level of achievement for one criterion will not necessarily attain low achievement levels for the other criteria. Teachers should not assume that the overall assessment of the students will produce any particular distribution of marks.• It is expected that the assessment criteria be made available to students.Internal assessment detailsMathematical explorationDuration: 10 teaching hoursWeighting: 20%IntroductionThe internally assessed component in this course is a mathematical exploration. This is a short report writtenby the student based on a topic chosen by him or her, and it should focus on the mathematics of that particulararea. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), withaccompanying commentary, good mathematical writing and thoughtful reflection. A student should develophis or her own focus, with the teacher providing feedback via, for example, discussion and interview. Thiswill allow the students to develop areas of interest to them without a time constraint as in an examination, andallow all students to experience a feeling of success.The final report should be approximately 6 to 12 pages long. It can be either word processed or handwritten.Students should be able to explain all stages of their work in such a way that demonstrates clear understanding.While there is no requirement that students present their work in class, it should be written in such a way thattheir peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sourcesneed to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.The purpose of the explorationThe aims of the mathematics HL course are carried through into the objectives that are formally assessed aspart of the course, through either written examination papers, or the exploration, or both. In addition to testingthe objectives of the course, the exploration is intended to provide students with opportunities to increase theirunderstanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics.These are noted in the aims of the course, in particular, aims 6–9 (applications, technology, moral, social66 Mathematics HL guide Internal assessmentand ethical implications, and the international dimension). It is intended that, by doing the exploration,students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. Itwill enable students to acquire the attributes of the IB learner profile.The specific purposes of the exploration are to:• develop students' personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics• provide opportunities for students to complete a piece of mathematical work over an extended period of time• enable students to experience the satisfaction of applying mathematical processes independently• provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics• encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool• enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work• provide opportunities for students to show, with confidence, how they have developed mathematically.Management of the explorationWork for the exploration should be incorporated into the course so that students are given the opportunity tolearn the skills needed. Time in class can therefore be used for general discussion of areas of study, as well asfamiliarizing students with the criteria. Further details on the development of the exploration are included inthe teacher support material.Requirements and recommendationsStudents can choose from a wide variety of activities, for example, modelling, investigations and applicationsof mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in theteacher support material. However, students are not restricted to this list.The exploration should not normally exceed 12 pages, including diagrams and graphs, but excluding thebibliography. However, it is the quality of the mathematical writing that is important, not the length.The teacher is expected to give appropriate guidance at all stages of the exploration by, for example, directingstudents into more productive routes of inquiry, making suggestions for suitable sources of information, andproviding advice on the content and clarity of the exploration in the writing-up stage.Teachers are responsible for indicating to students the existence of errors but should not explicitly correct theseerrors. It must be emphasized that students are expected to consult the teacher throughout the process.All students should be familiar with the requirements of the exploration and the criteria by which it is assessed.Students need to start planning their explorations as early as possible in the course. Deadlines should be firmlyestablished. There should be a date for submission of the exploration topic and a brief outline description, adate for the submission of the first draft and, of course, a date for completion.In developing their explorations, students should aim to make use of mathematics learned as part of the course.The mathematics used should be commensurate with the level of the course, that is, it should be similar to thatsuggested by the syllabus. It is not expected that students produce work that is outside the mathematics HLsyllabus—however, this is not penalized. Mathematics HL guide 67 Internal assessmentInternal assessment criteriaThe exploration is internally assessed by the teacher and externally moderated by the IB using assessmentcriteria that relate to the objectives for mathematics HL.Each exploration is assessed against the following five criteria. The final mark for each exploration is the sumof the scores for each criterion. The maximum possible final mark is 20.Students will not receive a grade for mathematics HL if they have not submitted an exploration. Criterion A Communication Criterion B Mathematical presentation Criterion C Personal engagement Criterion D Reflection Criterion E Use of mathematicsCriterion A: CommunicationThis criterion assesses the organization and coherence of the exploration. A well-organized explorationincludes an introduction, has a rationale (which includes explaining why this topic was chosen), describes theaim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow.Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached asappendices to the document. Achievement level Descriptor 0 The exploration does not reach the standard described by the descriptors below. 1 The exploration has some coherence. 2 The exploration has some coherence and shows some organization. 3 The exploration is coherent and well organized. 4 The exploration is coherent, well organized, concise and complete.Criterion B: Mathematical presentationThis criterion assesses to what extent the student is able to:• use appropriate mathematical language (notation, symbols, terminology)• define key terms, where required• use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate.Students are expected to use mathematical language when communicating mathematical ideas, reasoning andfindings.68 Mathematics HL guide Internal assessmentStudents are encouraged to choose and use appropriate ICT tools such as graphic display calculators,screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, toenhance mathematical communication. Achievement level Descriptor 0 The exploration does not reach the standard described by the descriptors below. 1 There is some appropriate mathematical presentation. 2 The mathematical presentation is mostly appropriate. 3 The mathematical presentation is appropriate throughout.Criterion C: Personal engagementThis criterion assesses the extent to which the student engages with the exploration and makes it their own.Personal engagement may be recognized in different attributes and skills. These include thinking independentlyand/or creatively, addressing personal interest and presenting mathematical ideas in their own way. Achievement level Descriptor 0 The exploration does not reach the standard described by the descriptors below. 1 There is evidence of limited or superficial personal engagement. 2 There is evidence of some personal engagement. 3 There is evidence of significant personal engagement. 4 There is abundant evidence of outstanding personal engagement.Criterion D: ReflectionThis criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflectionmay be seen in the conclusion to the exploration, it may also be found throughout the exploration. Achievement level Descriptor 0 The exploration does not reach the standard described by the descriptors below. 1 There is evidence of limited or superficial reflection. 2 There is evidence of meaningful reflection. 3 There is substantial evidence of critical reflection. Mathematics HL guide 69 Internal assessmentCriterion E: Use of mathematicsThis criterion assesses to what extent and how well students use mathematics in the exploration.Students are expected to produce work that is commensurate with the level of the course. The mathematicsexplored should either be part of the syllabus, or at a similar level or beyond. It should not be completelybased on mathematics listed in the prior learning. If the level of mathematics is not commensurate with thelevel of the course, a maximum of two marks can be awarded for this criterion.The mathematics can be regarded as correct even if there are occasional minor errors as long as they do notdetract from the flow of the mathematics or lead to an unreasonable outcome.Sophistication in mathematics may include understanding and use of challenging mathematical concepts,looking at a problem from different perspectives and seeing underlying structures to link different areas ofmathematics.Rigour involves clarity of logic and language when making mathematical arguments and calculations.Precise mathematics is error-free and uses an appropriate level of accuracy at all times. Achievement level Descriptor 0 The exploration does not reach the standard described by the descriptors below. 1 Some relevant mathematics is used. Limited understanding is demonstrated. 2 Some relevant mathematics is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated. 3 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Good knowledge and understanding are demonstrated. 4 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication expected. Good knowledge and understanding are demonstrated. 5 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated. 6 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated.70 Mathematics HL guide AppendicesGlossary of command termsCommand terms with definitionsStudents should be familiar with the following key terms and phrases used in examination questions, whichare to be understood as described below. Although these terms will be used in examination questions, otherterms may be used to direct students to present an argument in a specific way.Calculate Obtain a numerical answer showing the relevant stages in the working.Comment Give a judgment based on a given statement or result of a calculation.Compare Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.Compare and Give an account of the similarities and differences between two (or more) items orcontrast situations, referring to both (all) of them throughout.Construct Display information in a diagrammatic or logical form.Contrast Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.Deduce Reach a conclusion from the information given.Demonstrate Make clear by reasoning or evidence, illustrating with examples or practical application.Describe Give a detailed account.Determine Obtain the only possible answer.Differentiate Obtain the derivative of a function.Distinguish Make clear the differences between two or more concepts or items.Draw Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.Estimate Obtain an approximate value.Explain Give a detailed account, including reasons or causes.Find Obtain an answer, showing relevant stages in the working.Hence Use the preceding work to obtain the required result.Hence or otherwise It is suggested that the preceding work is used, but other methods could also receive credit.Identify Provide an answer from a number of possibilities. Mathematics HL guide 71 Glossary of command termsIntegrate Obtain the integral of a function.Interpret Use knowledge and understanding to recognize trends and draw conclusions from given information.Investigate Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.Justify Give valid reasons or evidence to support an answer or conclusion.Label Add labels to a diagram.List Give a sequence of brief answers with no explanation.Plot Mark the position of points on a diagram.Predict Give an expected result.Prove Use a sequence of logical steps to obtain the required result in a formal way.Show Give the steps in a calculation or derivation.Show that Obtain the required result (possibly using information given) without the formality of proof. "Show that" questions do not generally require the use of a calculator.Sketch Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.Solve Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.State Give a specific name, value or other brief answer without explanation or calculation.Suggest Propose a solution, hypothesis or other possible answer.Verify Provide evidence that validates the result.Write down Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown.72 Mathematics HL guide AppendicesNotation listOf the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of theInternational Organization for Standardization (ISO). This notation is used in the examination papers for this coursewithout explanation. If forms of notation other than those listed in this guide are used on a particular examinationpaper, they are defined within the question in which they appear.Because students are required to recognize, though not necessarily use, IB notation in examinations, it isrecommended that teachers introduce students to this notation at the earliest opportunity. Students are not allowedaccess to information about this notation in the examinations.Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3, ...} the set of integers, {0, ± 1, ± 2, ± 3, ...}+ the set of positive integers, {1, 2, 3, ...} the set of rational numbers+ the set of positive rational numbers, {x \| x ∈ , x > 0} the set of real numbers+ the set of positive real numbers, {x \| x ∈ , x > 0} the set of complex numbers, {a + ib \| a , b ∈ }i −1z a complex numberz∗ the complex conjugate of z z the modulus of zarg z the argument of zRe z the real part of zIm z the imaginary part of zcisθ cos θ + i sin θ{x1 , x2 , ...} the set with elements x1 , x2 , ...n( A) the number of elements in the finite set A{x \| } the set of all x such that∈ is an element of∉ is not an element of∅ the empty (null) setU the universal set∪ union Mathematics HL guide 73 Notation list∩ intersection⊂ is a proper subset of⊆ is a subset ofA′ the complement of the set AA× B the Cartesian product of sets A and B (that is, = {( a , b) a ∈ A , b ∈ B} ) A× Ba\|b a divides b 1 tha1/ n , n a a to the power of , n root of a (if a ≥ 0 then n a ≥0) n  x for x ≥ 0, x ∈ x the modulus or absolute value of x, that is  − x for x < 0, x ∈ ≡ identity≈ is approximately equal to> is greater than≥ is greater than or equal to< is less than≤ is less than or equal to>/ is not greater than</ is not less than⇒ implies⇐ is implied by⇔ implies and is implied by[ a , b] the closed interval a ≤ x ≤ b] a, b [ the open interval a < x < bun the n th term of a sequence or seriesd the common difference of an arithmetic sequencer the common ratio of a geometric sequenceSn the sum of the first n terms of a sequence, u1 + u2 + ... + unS∞ the sum to infinity of a sequence, u1 + u2 + ... n∑ui =1 i u1 + u2 + ... + un n∏u i =1 i u1 × u2 × ... × un74 Mathematics HL guide Notation list n n!  r !(n − r )!r f :A→ B f is a function under which each element of set A has an image in set Bf :x y f is a function under which x is mapped to yf ( x) the image of x under the function ff −1 the inverse function of the function ff g the composite function of f and glim f ( x) the limit of f ( x) as x tends to ax→ady the derivative of y with respect to xdxf ′( x) the derivative of f ( x) with respect to xd2 y the second derivative of y with respect to xdx 2f ′′( x) the second derivative of f ( x) with respect to xdn y th the n derivative of y with respect to xdx nf ( ) ( x) n th the n derivative of f ( x) with respect to x∫ y dx the indefinite integral of y with respect to x b∫a y dx the definite integral of y with respect to x between the limits x = a and x = bex the exponential function of xlog a x the logarithm to the base a of xln x the natural logarithm of x, log e xsin, cos, tan the circular functionsarcsin, arccos,   the inverse circular functionsarctan csc, sec, cot the reciprocal circular functionsA( x, y ) the point A in the plane with Cartesian coordinates x and y[ AB] the line segment with end points A and B Mathematics HL guide 75 Notation list AB [ the length of AB ] ( AB) the line containing points A and B Â the angle at A ˆ CAB the angle between [ CA ] and [ AB] ∆ABC the triangle whose vertices are A, B and Cv the vector v → the vector represented in magnitude and direction by the directed line segment AB from A to B →a the position vector OAi, j, k unit vectors in the directions of the Cartesian coordinate axes a the magnitude of a → →\|AB\| the magnitude of AB v⋅w the scalar product of v and w v×w the vector product of v and wI the identity matrix P(A) the probability of event A P( A′) the probability of the event "not A " P( A \| B ) the probability of the event A given B x1 , x2 , ... observations f1 , f 2 , ... frequencies with which the observations x1 , x2 , ... occur Px the probability distribution function P (X = x) of the discrete random variable X f ( x) the probability density function of the continuous random variable X F ( x) the cumulative distribution function of the continuous random variable X E(X ) the expected value of the random variable X Var ( X ) the variance of the random variable X µ population mean kσ 2 ∑ f (x i i − µ )2 k population variance, σ2 = i =1 n , where n = ∑f i =1 iσ population standard deviation76 Mathematics HL guide
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Description: WARNING!! If you were enrolled in the old PCAL 1A course, you must enroll in the old PCAL 1B course. Please contact a customer service representative to assist you. This precalculus course is designed to incorporate the two traditional courses of trigonometry and analytic geometry. As the name suggests, the precalculus course is intended to build upon existing algebra and geometry skills while preparing a student for the concepts he or she will be learning in a calculus course. It is important that a student has taken courses in algebra II and geometry before attempting to take precalculus. Precalculus 1B will primarily cover the trigonometry and analytic geometry portion, while the first semester course, Precalculus 1A, centered more on parent functions and transformations of their graphs, sequences and series, and limits. In this course, we use the concept of mathematical relations and functions to model, describe, and solve mathematical problems. Through the use of new functions defined in trigonometry (the study of triangles), the algebra and geometry involved in analytic geometry, and the aid of technology, you will learn how to become a more effective problem-solver. IMPORTANT: This course is different from any course you have enrolled in before. You will complete all of your assignments using a digital notepad system which is a recommended purchase for this course. We have tested and provided instructions for the ACECAD DigiMemo 692 or the ACECAD DigiMemo L2 for both PC (Windows XP/Vista/7) and Mac (OS X) users. If you choose to use a different system, please understand that you must be able to export the file(s) as a PDF for submission to your instructor. When you complete your assignment, you will transfer your assignment from the digital notepad system to your computer and then convert the assignment to a PDF file using the software that came with your system. To submit your assignment you will upload the PDF file using the appropriate upload page in Moodle. Please be aware technical support is available for the ACECAD DigiMemo 692 or the ACECAD DigiMemo L2 only: TTUISD staff is unable to offer technical support for scanners or multi-function scanning hardware and software. Please refer to the support documentation bundled with your particular device and/or seek direct support from its manufacturer. Recommended purchase Course: HIGH PRECA1B / Online Schedule Number: 8840 Instructor(s): Jeanine Howell Paul Vann Location: Dates: Units: 0.5 Academic Credits Lessons/Exams: 5
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Ideal for classroom, group, or individual study, our tutorials and targeted drills increase comprehension while enhancing math skills. Color icons and graphics throughout the book highlight practice problems, charts, and figures. REAs test-taking tips and strategies give students an added boost of confidence so they can succeed on the exam. The book contains three diagnostic tests that are perfect for classroom quizzes or homework, plus a full-length practice exam that allows students to test their knowledge and reinforce what theyve learned. Our interactive TestWare CD features the books tests with automatic scoring, diagnostic feedback, and onscreen detailed explanations of answers, allowing students to focus on areas in need of further study. Whether used in a classroom, for group study, or individual review, this book is a must for any Georgia student preparing for the CRCT math exam! For 50 years, REA has been helping students study smarter and excel on important exams. REAs test preps for state-required exams are teacher-recommended and written by experts who have mastered the test. Synopsis: Georgia CRCT (Criterion-Referenced Competency Tests) 8th Grade Mathematics with Practice Tests on CD Completely aligned with the Georgia Performance Standards! REA helps Grade 8 students get ready for CRCT math.This book provides the instruction and practice students need to attain full competency and pass the Georgia CRCT:
Multivariable Calculus: Concepts and Contexts 9781439046920 ISBN: 1439046921 Edition: Internatio Publisher: Brooks/Cole Summary: Stewart, James is the author of Multivariable Calculus: Concepts and Contexts, published under ISBN 9781439046920 and 1439046921. Two Multivariable Calculus: Concepts and Contexts textbooks are available for sale on ValoreBooks.com, and one used from the cheapest price of $26.65. Seller Rating:(0) Ships From:Lancaster, PAShipping:Standard, ExpeditedComments:ALTERNATE EDITION: Same content as the US edition, except that some exercises use the Metric system instead of the I... [more]ALTERNATE EDITION: Same content as the US edition, except that some exercises use the Metric system instead of the Imperial one. Like new condition due to very careful use, no markings or highlights. [less] ALTERNATE EDITION: Same content as the US edition, except that some exercises use the Metric system instead of the Imperial one. Like new condition due to very careful use, no markings or highli [more] ALTERNATE EDITION: Same content as the US edition, except that some exercises use the Metric system instead of the Imperial one. Like new condition due to very careful use, no markings or highlights.[less]
Mathematical Modeling Science and engineering students depend heavily on concepts of mathematical modeling. In an age where almost everything is done on a computer, author ...Show synopsisScience and engineering students depend heavily on concepts of mathematical modeling. In an age where almost everything is done on a computer, author Clive Dym believes that students need to understand and "own" the underlying mathematics that computers are doing on their behalf. His goal for Principles of Mathematical Modeling, Second Edition, is to engage the student reader in developing a foundational understanding of the subject that will serve them well into their careers. The first half of the book begins with a clearly defined set of modeling principles, and then introduces a set of foundational tools including dimensional analysis, scaling techniques, and approximation and validation techniques. The second half demonstrates the latest applications for these tools to a broad variety of subjects, including exponential growth and decay in fields ranging from biology to economics, traffic flow, free and forced vibration of mechanical and other systems, and optimization problems in biology, structures, and social decision making. Prospective students should have already completed courses in elementary algebra, trigonometry, and first-year calculus and have some familiarity with differential equations and basic physics. * Serves as an introductory text on the development and application of mathematical models * Focuses on techniques of particular interest to engineers, scientists, and others who model continuous systems * Offers more than 360 problems, providing ample opportunities for practice * Covers a wide range of interdisciplinary topics--from engineering to economics to the sciences * Uses straightforward language and explanations that make modeling easy to understand and apply New to this Edition: * A more systematic approach to mathematical modeling, outlining ten specific principles * Expanded and reorganized chapters that flow in an increasing level of complexity * Several new problems and updated applications * Expanded figure captions that provide more information * Improved accessibility and flexibility for teaching0122265518-5Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780122265518-4New. Brand New Original US Edition, Perfect Condition. Printed...New. Brand New Original US Edition, Perfect Condition. Printed in English. Excellent Quality, Service and customer satisfaction guaranteed! Description:Fine. Hardcover. Almost new condition. SKU: 9780122265518-2-0-3...Fine. Hardcover. Almost new condition. SKU: 9780122265518-2
Hercules StatisticsIt requires real understanding and good work over the entire school year. Math concepts build on each other so that future topics depend on understanding previous material. Therefore, misunderstanding one topic can cause continuous problems down the road
Precalculus Agile Mind Precalculus deepens students' understanding of the behavior of functional relationships and builds conceptual knowledge and skills essential to success in advanced mathematics and college coursework. This Precalculus course is one in which students: Use functions, equations, sequences, series, vectors, and limits as tools to express generalizations and to analyze and understand a variety of mathematical relationships and real-world phenomena Expand and develop their use of functions and their properties to choose appropriate models for real-world problem situations to answer meaningful questions Build on and expand their experiences with functions from Algebra I, Algebra II, and Geometry as they continue to explore the characteristics and behavior of functions (including rate of change and limits), and the most important families of functions that model real-world phenomena, especially transcendental functions Extend their work in functions, including polynomial, rational, radical, exponential, power, logarithmic, and trigonometric functions Continue to work with operations on functions, including composition of functions Talk with us about working together to close the achievement gaps in math and science. For all students. Preparation for high stakes assessments with both automatically graded and open-response questions Real-time reporting of progress that allows students to take responsibility for their own learning Robust Supports for Teachers In addition to Internet-delivered services, educators and administrators also receive face-to-face seminars, mentoring, and high-quality support materials to help them manage their demanding workloads, enhance their expertise, and dramatically improve outcomes for their students. Online and face-to-face professional development seminars and mentoring directly tied to practice Day-by-day lesson support with advice and classroom strategies, equipping teachers to enact each day of instruction to achieve success for all students Collaboration with Leading Educational Research Center Our mathematics and academic youth programs are developed in collaboration with the Charles A. Dana Center at The University of Texas at Austin. Working with leading educators throughout the country, we have developed mathematics programs for middle school, Algebra I, Geometry, Algebra II, Precalculus, Calculus, and AP Statistics. These programs build on the Dana Center's trusted work with more than 100,000 teachers over the past decade and on the core belief that all students can succeed in mathematics if given the opportunity.
THE PRINCIPLES OF MATHEMATICS. 451. I wish here emphatically to remind you that both in Geometry and in Algebra a clear grasp of these general ideas is not ... File Type:PDF in this pathway on the applications of mathematics, one of the primary purposes of Principles of Mathematics will be to develop ... File Type:PDF Principles and Standards for School Mathematics is intended to ..... The vision of school mathematics in Principles and Standards is based on students' ... File Type:PDF Abstract: Zoltan Dienes' principles of mathematical learning have been an integral part of mathematics education literature and applied both to the teaching ... File Type:PDF The principle of permutation, in mathematics, is about arranging a group of numbers or .... Hence the principle of "combination" in mathematics applies. By ... File Type:PDF
Mathematical Ideas (12th Edition) 9780321693815 ISBN: 0321693817 Edition: 12 Pub Date: 2011 Publisher: Addison Wesley Summary: Mathematical Ideas offers students a comprehensive understanding of how they can relate math to everyday situations and even more unique situations such as those from film and television. It uses an innovative approach to guide students through the complex mathematical concepts through relatively easy to understand approaches that are easy to apply. These methods form part of a very readable and accessible textbook. ...It also offers excellent study tools to aid subject comprehension. We offer many mathematics textbooks of this calibre to buy brand new or to rent in good condition. We also offer a buyback service for those with used textbooks to sell. Miller, Charles David is the author of Mathematical Ideas (12th Edition), published 2011 under ISBN 9780321693815 and 0321693817. Seven hundred ninety seven Mathematical Ideas (12th Edition) textbooks are available for sale on ValoreBooks.com, three hundred ninety three used from the cheapest price of $45.30, or buy new starting at $110.80 12th edition. WHY PAY MORE? Annotated Instructor's Edition -- same content and pagination as student version but with extra answers and/or teaching tips in the margins. Includes Movie problem supplements at end of book. Light internal markings (minor). Moderate to Fair amount of cover wear. Some Areas with dogeared pages, and Some page creases at corners and along edges. Bookstore sticker on spine. Nice for the price!! Enjoy!!1694065. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less] I used this book for a mathematical course for liberal arts degree. It was the second part and the course number was 152. We studied line graphs, geometry, statistics, and probability. I felt that the material was well described both in the book and in class, the online tutorials were overkill. I felt that this book only partially prepared me for the GRE exam, I did study for the test using this particular text, and I did complete the practice quizzes, but I did not do quite as well as I had hoped. Overall I thought that it prepared pretty well, but could use some improvement in the English and Math sections.
What is ap calculus? Answer AP calculus is a class that is offered in many high schools and secondary schools for advanced students. The term 'AP' stands for advanced placement. Often times these AP classes also offer college credit as they are considered to be at the college level
NumericalMethods in Engineering with MATLAB® is a text for engineering students and a reference for practicing engineers. The choice of numericalmethods was based on their relevance to engineering problems. Every method is discussed thoroughly and illustrated with problems involving both hand computation and programming. MATLAB M-files accompany each method and are available on the book website. This code is made simple and easy to understand by avoiding complex book-keeping schemes, while maintaining the essential features of the method. This book presents an introduction to MATLAB and its applications in engineering problem solving. The book is designed as an introductory course in MATLAB for engineers. The classical methods of electrical circuits, control systems, numericalmethods, optimization, direct numerical integration methods, engineering mechanics and mechanical vibrations are covered using MATLAB software. This is a unique monograph on numerical conformal mapping that gives a comprehensive account of the theoretical, computational and application aspects of the problems of determining conformal modules of quadrilaterals and of mapping conformally onto a rectangle. It contains a detailed study of the theory and application of a domain decomposition method for computing the modules and associated conformal mappings of elongated quadrilaterals, of the type that occur in engineering applications. This book will interest researchers, scientists, engineers and graduate students in many disciplines, who make use of mathematical modeling and computer simulation. Although it represents only a small sample of the research activity on numerical simulations, the book will certainly serve as a valuable tool for researchers interested in getting involved in this multidisciplinary field. It will be useful to encourage further experimental and theoretical researches in the above mentioned areas of numerical simulation. Since the original publication of this book, available computer power has increased greatly. Today, scientific computing is playing an ever more prominent role as a tool in scientific discovery and engineering analysis. In this second edition, the key addition is an introduction to the finite element method. This is a widely used technique for solving partial differential equations (PDEs) in complex domains. This text introduces numericalmethods and shows how to develop, analyze, and use them. STATISTICAL METHODS FOR ENGINEERS offers a balanced, streamlined one-semester introduction to Engineering Statistics that emphasizes the statistical tools most needed by practicing engineers. Using real engineering problems with real data based on actual journals and consulting experience in the field, users see how statistics fits within the methods of engineering problem solving. The book teaches users how to think like an engineer at analyzing real data and planning a project the same way they will in their careers. Written as both a textbook and a handy reference, this text deliberately avoids complex mathematics assuming only basic familiarity with geodynamic theory and calculus. Here, the authors have brought together the key numerical techniques for geodynamic modeling, demonstrations of how to solve problems including lithospheric deformation, mantle convection and the geodynamo. Building from a discussion of the fundamental principles of mathematical and numerical modeling, the text moves into critical examinations of each of the different techniques before concluding with a detailed analysis of specific geodynamic applications
As in the first volume, More Fallacies, Flaws, and Flimflam contains items ranging from howlers (outlandish procedures that nonetheless lead to a correct answer) to deep or subtle errors often made by strong students. Although some are provided for entertainment, others challenge the reader to determine exactly where things go wrong. Items are sorted by subject matter. Elementary teachers will find chapter 1 of most use, while middle and high schoolteachers will find chapters 1, 2, 3, 7, and 8 applicable to their levels. College instructors can delve for material in every part of the book. There are frequent references to The College Mathematics Journal; these are denoted by CMJ. Excerpt: 5.8 A standard box problem (p.89) Dale R. Buske put on a recent calculus examination the following standard problem: Problem A crate open at the top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the crate is to be constructed so as to have minimal surface area, find its dimensions. One student started with this formula for the surface area of the crate: SA = 4x + 4y, where x was the length of one side of the base and y was the height of the crate. (After all, there are four line segments of length x on the bottom of the crate, four line segments of length y on the sides, and the four line segments on top do not count since the crate has an open top.) The student then correctly went on to use the volume constraint 4 = x2y to find y = 4/x2 and arrive at the formula SA = 4x + 16/x2. Taking the derivative and applying the condition for a maximum leads to the correct answer: the crate should have a base with dimensions 2 meters by 2 meters and a height of 1 meter. About the Author Ed Barbeau graduated from the University of Toronto (BA, MA) and received his doctorate from the University of Newcastle-upon-Tyne, England in 1964. He was a Postdoctoral Fellow at Yale University in 1966-67, and taught at the University of Toronto from 1967 until 2003. He is currently retired. Barbeau was named Fellow of the Ontario Institute for Studies in Education (1989), and has received the David Hilbert Award (1991) from the World Federation of National Mathematics Competitions and the Adrien Pouliot Award (1995) from the Canadian Mathematical Society. He has published a number of books with the MAA as well as two books, Polynomials and Pell's Equation with Springer. He has been invited to give talks frequently to groups of teachers, students and the general public, and has presented three radio broadcasts, "Proof and truth in mathematics," in the Canadian Broadcasting Corporation series, Ideas.
is designed for the sophomore/junior level introduction to discrete mathematics taken by students preparing for future coursework in areas such as math,computer science and engineering. Rosen has become a bestseller largely due to how effectively it addresses the main portion of the discrete market,which is typically characterized as the mid to upper level in rigor. The strength of Rosen's approach has been the effective balance of theory with relevant applications,as well as the overall comprehensive nature of the topicondrath Posted January 25, 2012 NOT Recommended (if you have a choice) This book was required for my Discrete Mathematics course. Unfortunately the way the material is presented is difficult to understand. I found myself looking for other resources to help me on topics I was stuck on. While I don't doubt the author's expertise, I had a hard time interpreting methods in which to solve problems in the chapter reviews, as if there was a piece of the puzzle always missing. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Discrete Mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Throughout the text, engaging anecdotes and topics of interest inform as well as motivate learners. The text is ideal for one- or two-semester courses and for students who are typically mathematics, mathematics education, or computer science majors. Part I teaches student how to write proofs; Part II focuses on computation and problem solving. The second half of the book may also be suitable for introductory courses in combinatorics and graphDiscrete Mathematics
KEY MESSAGE: El KEY TOPICS: Real Numbers And Algebraic Expressions; Equations, Inequalities, And Problem Solving; Graphs and Functions; Systems of Linear Equations and Inequalities; Exponents, Polynomials, and Polynomial Functions; Rational Expressions; Rational Exponents, Radicals, and Complex Numbers; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conic Sections; Sequences, Series, and the Binomial Theorem MARKET: For all readers interested in intermediate algebra, and for all readers learning or revisiting essential skills in intermediate algebra through the use of lively and up-to-date applications. This book is great for students planning on continuing on to College Algebra. This book explains things well. It has practice sections and a very helpful CD. KEY MESSAGE: El For all readers interested in intermediate algebra, and for all readers learning or revisiting essential skills in intermediate algebra through the use of lively and up-to-date applications. More Reviews and Recommendations Biography: An award-winning instructor and best-selling author, Elayn Martin-Gay has taught mathematics at the University of New Orleans for more than 25 years. Her numerous teaching awards include the local University Alumni Association's Award for Excellence in Teaching, and Outstanding Developmental Educator at University of New Orleans, presented by the Louisiana Association of Developmental Educators. Prior to writing textbooks, Elayn developed an acclaimed series of lecture videos to support developmental mathematics students in their quest for success. These highly successful videos originally served as the foundation material for her texts. Today, the videos are specific to each book in the Martin-Gay series. Elayn also pioneered the Chapter Test Prep Video to help students as they prepare for a test–their most "teachable moment!" Elayn's experience has made her aware of how busy instructors are and what a difference quality support makes. For this reason, she created the Instructor-to-Instructor video series. These videos provide instructors with suggestions for presenting specific math topics and concepts in basic mathematics, Prealgebra, beginning algebra, and intermediate algebra. Seasoned instructors can use them as a source for alternate approaches in the classroom. New or adjunct faculty may find the CDs useful for review. Her textbooks and acclaimed video program support Elayn's passion of helping every student to succeed. Great textbook! I purchased this book for an intermediate algebra class I am taking. The book is very helpful in diagramming each step of the problem. There is also on-line videos to accompany each chapter where the books author actually teaches you the lesson. A great tool for brushing up your algebra skills before taking a college algebra class.
Schaum's has Satisfied Students for 50 Years. Now Schaum's Biggest Sellers are in New Editions! For half a century, more than 40 million students have trusted Schaum's to help them study faster, learn better, and get top grades. Now Schaum's celebrates its 50th birthday with a brand-new look, a new format with hundreds of practice problems,... more... This new-in-paperback edition provides an introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. Clear explanations of both theory and applications, and almost 600 exercises are included in the text. - ;This new-in-paperback... more... Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of... more... Students and researchers in physics, engineering and other sciences will find this compilation of one letter abbreviations used in mathematics and physics invaluable. All the information included is practical, rarely used results are excluded. Excellent to keep as a handy reference! Selected abbreviations you will find inside this guide: A - Ampere,... more... Students and research workers in mathematics, physics, engineering and other sciences will find this compilation of Logarithmic Identities invaluable. All the information included is practical, rarely used results are excluded. Great care has been taken to present all results concisely and clearly. Excellent to keep as a handy reference! If you don't... more...... more... Comprises of articles that were offered as a tribute to one of the world's greatest mathematician, Alexander Grothendieck. This book carries contributions that contain material that is considered foundational to the subject. It carries topics that are addressed by top-notch contributors to match the breadth of Grothendieck's own interests. more... This volume is a homage to the memory of the Spanish mathematician Federico Gaeta (1923-2007). Apart from a historical presentation of his life and interaction with the classical Italian school of algebraic geometry, the volume presents surveys and original research papers on the mathematics he studied. Specifically, it is divided into three parts:... more...
Mathematics for Physics and Physicists Walter Appel What can a physicist gain by studying mathematics? By gathering together everything a physicist needs to know about mathematics in one comprehensive and accessible guide, this is the question Mathematics for Physics and Physicists successfully takes on. The author, Walter Appel, is a renowned mathematics educator hailing from one of the best schools of France's prestigious Grandes Écoles, where he has taught some of his country's leading scientists and engineers. In this unique book, oriented specifically toward physicists, Appel shows graduate students and researchers the vital benefits of integrating mathematics into their study and experience of the physical world. His approach is mathematically rigorous yet refreshingly straightforward, teaching all the math a physicist needs to know above the undergraduate level. Appel details numerous topics from the frontiers of modern physics and mathematics--such as convergence, Green functions, complex analysis, Fourier series and Fourier transform, tensors, and probability theory--consistently partnering clear explanations with cogent examples. For every mathematical concept presented, the relevant physical application is discussed, and exercises are provided to help readers quickly familiarize themselves with a wide array of mathematical tools. Mathematics for Physics and Physicists is the resource today's physicists must have to strengthen their math skills and to gain otherwise unattainable insights into their fields of study "The majority of applied mathematical fields presently require so much specialization that mathematics often takes a back seat to the particular field of study. This book not only contains a great deal of the mathematics necessary to seriously study physics but also encourages physicists and potential physicists to embrace mathematics."--J.T. Zerger, Choice "Walter Appel, a theoretical physicist and mathematics educator who currently teaches mathematics at the Henri Poincaré School in France, seeks in his book--appearing here in translation--to cover all the mathematics that a physicist requires above undergraduate level, including recent topics such as convergence, Green functions and Fourier series, as well as offering biographical sketches of mathematicians and problem sets."--Times Higher Education "There is a law, if not a physical law, that ensures that whenever one is using a standard mathematical technique for a physical problem it is always the special case or a first principles argument that is required. Nothing is straightforward! For such a situation this book is ideal. It presents clear definitions and the rationale for such definitions. The style of the book is very readable and an interesting biographical asides of the mathematicians associated with the topics provide light relief from the depth of the the analysis. The book is both a valuable reference book and is a good pedagogic treatment of mathematical physics. It is a book that should be on many bookshelves."--Brian L. Burrows, Zentralblatt Math
Beginning Algebra (11th Edition) 9780321673480 ISBN: 0321673484 Edition: 11 Pub Date: 2010 Publisher: Addison Wesley Summary: Lial, Margaret L. is the author of Beginning Algebra (11th Edition), published 2010 under ISBN 9780321673480 and 0321673484. One thousand one Beginning Algebra (11th Edition) textbooks are available for sale on ValoreBooks.com, two hundred fifty two used from the cheapest price of $101.15, or buy new starting at $1671673480-3-0-3 Orders ship the same or next business day... [more]Beginning Algebra by Margaret Lial, John Hornsby, Terry McGinnis' c... [more]The Boundless alternative to 'Beginning Algebra by Margaret Lial, John Horns[lessBeginning Algebra by Margaret Lial, John Hornsby, Terry McGinnis' contains all the information you need for you to ace your class, including quiz [more] DIGITAL ALTERNATIVE: The Boundless alternative to 'Beginning Algebra by Margaret Lial, John Horns
Mathematics Advanced EPMATH125 Mathematics Advanced prepares students for undergraduate courses requiring a background in mathematics. The depth and treatment of this course is similar to 2 Unit Higher School Certificate mathematics. Topics include calculus (differentiation and integration), applications to the physical world and to trigonometry, exponential and logarithmic functions, growth and decay, probability and the binomial theorem. Available in 2014 In this course students will: 1.Obtain a background in topics mentioned in the brief course description. 2.Develop students' mathematical skills equivalent to 2 Unit HSC level. 3.Demonstrate a broad understanding of the course suitable for the commencement of undergraduate degree courses requiring a high level of mathematical facility. Content The course covers: 1.Sequences and Series: arithmetic and geometric sequences, general term, sum to n terms and limiting sum. 2.Introductory Calculus: stationary points, curve sketching, problems on maxima and minima, continuity and limits, the second derivative and points of inflection. 3.Integral calculus, primitives, the area under a curve, definite and indefinite integrals, volumes of solids of revolution. Approximate area under a curve - Trapezoidal and Simpson's Rule. 4.The calculus of exponential and logarithmic functions. Differentiation and integration related to exponential and logarithmic functions. 5. Trigonometric functions: radian measure, differentiation and integration related to trigonometric functions; curve sketching. 6. Calculus in the physical world - rates of change, displacement, velocity and acceleration. 7.Exponential growth and decay. 8.Probability: Venn diagrams, tree diagrams, counting techniques, permutations and combinations, the binomial probability distribution. Replacing Course(s) NA Transition NA Industrial Experience 0 Assumed Knowledge At least a Pass grade in EPMATH124 Mathematics Advanced. Modes of Delivery Internal Mode Teaching Methods Problem Based Learning Lecture Tutorial Assessment Items Examination: Class Progressive assessment consisting of class tests, 50%. Examination: Formal Final examination 50%. The duration of the exam will be two hours. Assess ability to solve problems and integrate various key concepts. Contact Hours Lecture: for 2 hour(s) per Week for 12 weeks Tutorial: for 2 hour(s) per Week for 12 weeks
Texas Instruments TI-36X Pro \| User Manual - Page 3 ...followed by instructions for keystroke examples that demonstrate the TI-36X Pro functions. Examples assume all default settings, as shown in the Modes section. Some screen...with other instructions. The answers are displayed on the Home screen. The TI-36X Pro screen can display a maximum of four... Texas Instruments TI-36X Pro \| User Manual - Page 5 ... activate the secondary function of a given key. Notice that 2ND appears as an indicator on the screen. To cancel it before entering data, ... gradians. NORM SCI ENG Sets the numeric notation mode. Numeric notation modes affect only the display of results, and not the accuracy of the values stored in ... Texas Instruments TI-36X Pro \| User Manual - Page 6 ...power is always a multiple of 3. Note: E is a shortcut key to enter a number in scientific notation format. The result displays in the numeric notation format selected in the mode menu. FLOAT...(To enter hex digits A through F, use and so on.) BIN binary OCT octal CLASSIC MATHPRINT CLASSIC... Texas Instruments TI-36X Pro \| User Manual - Page 8 ...hyperbolic functions sinh and sinh/. Press the key repeatedly to display the function that you want to enter. Multi-tap keys include z, X, Y, ... of this guidebook describe how to use the keys. Menus Menus give you access to a large number of calculator functions. Some menu keys, such as % h, display ... Texas Instruments TI-36X Pro \| User Manual - Page 9 ... and history !"#$ Press ! or " to move the cursor within an expression that you are entering or editing. Press % ! or % " to move the cursor directly to the beginning or..., the expression and its result are added automatically to the history. Use # and $ to scroll through the history. You can reuse a ... Texas Instruments TI-36X Pro \| User Manual - Page 10 ...The last entry performed on the home screen is stored to the variable ans. This variable is retained in memory, even after the calculator is turned off. To recall the value of ans: • Press % i (ans displays on the screen), or • Press any operations key ... Texas Instruments TI-36X Pro \| User Manual - Page 14 ...numerator. To enter fractions with operators or radicals, press P before you enter a number (in MathPrint™ mode only). • In MathPrint™ mode, press $... denominator or a lower limit, places the cursor in the history. Pressing enter will then paste the expression back to that MathPrint™ level. - ... Texas Instruments TI-36X Pro \| User Manual - Page 17 Powers, roots and inverses F Calculates the square of a value. The TI-36X Pro calculator evaluates expressions entered with F and a from left to right in both Classic and MathPrint™ modes. Raises a value to the power indicated. Use " ... Texas Instruments TI-36X Pro \| User Manual - Page 27 Logarithm and exponential functions D C (multi-tap keys) D yields the logarithm of a number to the base e (e ≈ 2.718281828459). D D ... with respect to variable, given the value at which to calculate the derivative and H (if not specified, the default is 1EM3). This function is valid only for real ... Texas Instruments TI-36X Pro \| User Manual - Page 28 ...investigations. You can use nDeriv( once in expression. Because of the method used to calculate nDeriv(, the calculator can return a false derivative value at a nondifferentiable point. ³ Problem Find the slope of the tangent line to the curve ... Texas Instruments TI-36X Pro \| User Manual - Page 29 ... U 4 z "" 2P%b3 < 2 - is zero. A maximum or The slope of the tangent line at x = -----3 minimum of the function must be at this point! Numeric integral %Q % Q calculates the numeric function integral of an expression with respect to a variable x, given a lower limit ... Texas Instruments TI-36X Pro \| User Manual - Page 32 ... the current entry, but the value assigned to the variable is used to evaluate the expression. To enter two or more variables in succession, press " after each. %h recalls the values of ... %{ and select 1: Yes to clear all variable values. Examples Start with %sclear screen Clear Var %{ ... Texas Instruments TI-36X Pro \| User Manual - Page 37 ... the model equation y=ax+b to the data using a least-squares fit. It displays values for a (slope) and b ...model equation y=a+b ln(x) to the data using a least squares fit and transformed values ln(x) and y. ...model equation y=axb to the data using a least-squares fit and transformed values ln(x) and ln... Texas Instruments TI-36X Pro \| User Manual - Page 39 ... integer (SINGLE) or a list of integers (LIST). The probability density function (pdf) is: Computes a cumulative probability at x for the discrete Poisson distribution ... are stored in the StatVars menu until the next statistics or regression calculation. The results must be interpreted based on which... Texas Instruments TI-36X Pro \| User Manual - Page 51 ... of the set of points of this function. To search closer to x = 18, change the Step value to smaller and ... closer to (18, 324). ³ Problem A charity collected $3,600 to help support a local food... and press Texas Instruments TI-36X Pro \| User Manual - Page 66 Conversions The CONVERSIONS menu permits you to perform a total of 20 conversions (or 40 if converting both ways). To access the CONVERSIONS menu, press % -. Press one of the numbers (1-5) to select, or press # and $ to scroll through and select one of the CONVERSIONS submenus. The submenus include... Texas Instruments TI-36X Pro \| User Manual - Page 71 ... may encounter. To correct the error, note the error type and determine the cause of the error. If you cannot recognize the error, ... is returned when you input an invalid value for area invNormal. ARGUMENT - This error is returned if: • a function does not have the correct number of arguments. • ... Texas Instruments TI-36X Pro \| User Manual - Page 72 ... get the COMPLEX error. DATA TYPE - You entered a value or variable that is the wrong data type. • For a function (including implied multiplication) or an instruction, you entered an argument that is an invalid data type, such as a complex number where... Texas Instruments TI-36X Pro \| User Manual - Page 77 ...Special handling may apply. See In case of difficulty Review instructions to be certain calculations were performed properly. Check the battery to ensure that it is fresh and properly installed. Change... Texas Instruments TI-36X Pro \| User Manual - Page 78 Texas Instruments Support and Service For general information Home Page: education.ti.com KnowledgeBase and education.ti.com/support e-..., Puerto Rico and Virgin Islands: Always contact Texas Instruments Customer Support before returning a product for service. All other customers: Refer to the ... Type your new search above The manual viewer requires the flash plugin to be installed and enabled. To view this page ensure that Adobe Flash Player version 10 or greater is installed.
Mathematics is the logical study of space, number, quantity and their relationships. It includes calculus, geometry, topology, number theory, probability and statistics, and many other subfields. This guide lists some of the most important reference books and databases available through this library. It is not complete; please don't hesitate to ask a librarian for suggestions if these works don't answer your needA collection of more than 750,000 citations and abstracts from major publishers in computing. The "advanced search" option is only available to ACM members, but the basic search and browse feature provide search options. A comprehensive index, with abstracts, to the research literature of mathematics. Corresponds to the print indexes Mathematical Reviews (MR) and Current Mathematical Publications (CMP). Published by the American Mathematical SocietyCompanion Encyclopedia of the History and Philosophy of Mathematical Sciences Offers lengthy, well-documented articles on the historical basis of mathematics and the cultural production of mathematical understanding. • Location(s): Ref QA 21 .C645 1994 Encyclopedia of Computers and Computer History Provides entries of people, organizations, and developments in the history of computing, "from the abacus to eBay." • Location(s): Ref QA 76.15 .E53 2001 Encyclopedia of Statistical Sciences An in-depth exploration of all fields of inquiry in which statistical methods are used. Includes articles on statistical terminology as well as on fields in which statistics are used. • Location(s): Ref QA 276.14 .E5 1982 Encyclopedic Dictionary of Mathematics Includes detailed discussions of mathematical concepts and excellent bibliographies. The appendices include tables of formulas and indexes by author, name, and subject. • Location(s): Ref QA 5 .I8313 1987 Offers over 6,000 definitions relating to computer applications as well as the hardware, software, and telecommunications behind these applications. The dictionary also addresses the concepts and theories of computing and associated fields. Covers thousands of terms related to the Web, e-commerce, security, and the technical and organizational infrastructure of the Internet. The World of Mathematics: A Small Library of the Literature of Mathematics from A'h-mose the Scribe to Albert Einstein This gem can sometimes provide answers to the most arcane of questions, while conducting a tour through the history and traditions of mathematics. It is dated, so won't cover recent theory, but has excellent background material and is well indexed. Shelved in the stacks. • Location(s): QA 3 .N48 Biographical sources Biographical Dictionary of Mathematicians Offers lengthy biographies of mathematicians from earliest times to the 20th century, including discussion of their work in the field. Each entry is followed by a detailed bibliography of primary and secondary sources. • Location(s): Ref QA 28 .B534 1991 Notable Matematicians from Ancient Times to the Present Provides biographical sketches of some 300 mathematicians from all time periods, including 50 women. Each biography includes selected writings and recommended readings. A timeline, list of mathematical prizes and their recipients, and a selective bibliography are included. • Location(s): Ref QA 28 .N66 1998 Women of Mathematics: A Biobibliographical Sourcebook Provides lengthy biographies and discussions of the contributions of women mathematicians with an emphasis on early times. Each biography is followed by a list of works by and about the subject. • Location(s): Ref QA 28 .W66 1987 mathematics and computer science. HA Statistics--general and census compilations Q Science (general) QA Mathematics 9-10 Mathematical logic 76 Computer science 101-141 Elementary mathematics. Arithmetic 150-271 Algebra, including machine theory, game theory 273-274 Probabilities 276-280 Mathematical statistics 297-299 Numerical analysis 299.8-433 Analysis 440-699 Geometry 611-614 Topology 801-939 Analytical mechanics T Technology TK 5101-6720 Telecommunication TK 7800-8360 Electronics, including computer hardware Journals Print journals are housed in the periodicals collection on the first floor of the library. Part of the arXiv.org preprint server, a moderated archive for research in several scientific disciplines, and sponsored by the Association for Computing Machinery, this site offers free access to thousands of technical papers in the field. From the School of Mathematics and Statistics at St. Andrews University, Scotland, this Web site provides biographies, historical overviews of topics, the history of mathematics in different cultures, and a "famous curves" page with illustrations, formulas, and other information.
Building on the success of its first three editions, the Fourth Edition of this market-leading text covers the important principles and real-world applications of plane geometry, with additional chapters on solid geometry, analytic geometry, and an introduction to trigonometry. Strongly influenced by both NCTM and AMATYC standards, the text takes an inductive approach that includes integrated activities and tools to promote hands-on application and discovery. New! Tables provide visual connections between figures and concepts and help students better assess their level of mastery and test readiness. New! Chapter Tests have been added to the end of every chapter. New! Proofs have been varied to include written and visual proofs, as well as comparisons, to support students with different learning styles. New! Exercise sets in the Student Study Guide, with cross-references to the text, offer additional practice and review. New! Technology-related margin features encourage the use of the Geometer's Sketchpad, graphing calculators, and further explorations. Highly visual approach begins with the presentation of an idea, followed by the examination and development of a theory, verification of the theory through deduction, and finally, application of the principles to the real world. Discovery features reinforce the text's inductive approach:activities integrated throughout enable students to discover geometry concepts on their own, and section tools provide with hands-on application of geometric concepts Applications reinforce the connection of geometry to the real world: high-interest Chapter Openers introduce the principal notion of the chapter and relate to the real world and A Perspective On... sections conclude each chapter, providing sketches that are interesting, sometimes historical, and always informative. Summaries of constructions, postulates, and theorems are provided, and an easy-to-navigate numbering system for postulates and theorems provides a user-friendly structure. In response to user feedback, paragraph proofs feature more prominently in this edition. Comprehensive appendices include Algebra Review and An Introduction to Logic. A glossary of terms, a summary of applications in the text, and selected answers are also provided in the back of the text. Related Subjects Meet the Author Daniel C. Alexander has taught mathematics at Parkland College in Champagne, Illinois, for the past fifteen years. Prior to his arrival at Parkland, Professor Alexander taught at the high-school level. He received his undergraduate and graduate degrees at Southern Illinois, Carbondale. Geralyn Koeberlein teaches mathematics at Mahomet-Seymour High School in Champagne, Illinois. She has been awarded several outstanding teacher awards throughout
Basic Math for Process Control Look Inside About A practical tutorial on the mathematics essential to the process control field, written by an experienced process control engineer for practicing engineers and students. A quick-and-easy review of the mathematics common to the field, including chapters on frequency response analysis, transfer functions and block diagrams, and the Z-N approximation. A handy desk reference for process control engineers - a helpful aid to students in mathematics courses.
Larry Feldman Get six applications for the price of two and ace your math classes. Rated 5/5 by AppCraver: Rated 5/5 by AppleBitesize: TheAppleBites.com: "High School Math becomes easy with Math Pro." Read the... If you are a math student or math teacher, "Two Points" will be one of your best friends. Business Analysts, Engineers, Architects, and other professionals will also consider this application to be essential. The concept is simple, and the execution is elegant. Simply enter the X and Y... Would you like a math ACT score in the 30It's good to be the curve wrecker. If you're in Algebra I or II, this application will help you beyond your highest expectations. Rated 5/5 by iear.org (I Education Apps Review). Read the full review here: Would you like a math SAT score in the 700Or, compute the value on a bell curve given the area beneath it.... Raise your hand if you think the only solutions to sin(x) = 0.5 are 30 degrees or pi/6 radians. Feel free to use your calculator. If you raised your hand, promptly use your hand to download this application. There are actually an infinite number of solutions to most inverse trigonometry... Would you like to ace your next test on quadratic functions? "Parabola" tells you everything you ever needed to know about parabolas but were afraid to ask. Enter a quadratic function in standard, vertex, or intercept form, and this application instantly displays: - The orientation... Pop quiz: you flip a fair coin 50 times. What is the probability that it lands on heads at least 25 times? The number of computations involved in answering that question is staggering. The formulas can cause nightmares. "Probability Pro" will tell you that the answer is 55.61%... You're taking Finite Mathematics, and you don't have this app? Give yourself an edge! "Finite Math Pro" contains a rich collection of tutorials, examples, and solvers for the following topics: Points and Lines: - Solvers and tutorials to find slopes, intercepts, equations... Calculus has been known to bring students to tears. Now you have an expert in your corner. This application contains a rich collection of examples, tutorials and solvers, crafted by a professional math tutor with over 20 years of applied mathematics and teaching experience. "Calculus... Whether you're looking to join the workforce or make a move to advance your career, let Digital Staffing help you land your dream job. Digital Staffing, a leader in IT staffing since 1996, is pleased to bring you our "Job Hunting" app to help you locate jobs, optimize your resume,... The honor roll is a good place to be. Let "Geometry Pro" serve as your portable tutor. This application contains a rich collection of examples, tutorials and solvers, crafted by a professional math tutor with over 20 years of applied mathematics and teaching experience. This...
What is Mathematica? Mathematica is the world's most powerful global computing environment. Ideal for use in engineering, mathematics, finance, physics, chemistry, biology, and a wide range of other fields, it makes possible a new level of automation in algorithmic computation, interactive manipulation, and dynamic presentation--as well as a whole new way of interacting with the world of data. Getting Mathematica... Mathematica is currently installed in the following locations: College of Health and Sciences Lab Mathematica can also be installed on: Faculty/staff school-owned machines: Installers are available from Bruce Weems Students' personally-owned machines: Students can buy discounted licenses through Wolfram's Web store, but if you're teaching with Mathematica or lots of students will be purchasing licenses, please contact Andy Dorsett for better discounts
arshbarger book reviews Student Solutions Manual for Harshbarger/Reynolds' Mathematical Applications for the Management, Life, and Social Sciences, 10th Solutions to Harshbarger The solutions are fine, the bad side is: still only the uneven numbered questions are solved. Why provide so many questions to students and no answers? My students (and especially the ones not "in
Book summary This book fills the gap between volumes on wavelets that are either too advanced (in terms of mathematical background required) or that contain too little mathematical theory underlying wavelets. It presents most of the theory underlying Fourier analysis and wavelets in a clear and comprehensive fashion-- without requiring advanced background in real analysis. Provides a careful balance between theory and practical algorithms, and features a clear presentation of applications to digital signal processing--e.g., data compression, digital filtering and singularity detection. Includes illustrations and MATLAB code used in many of the algorithms. Inner Product Spaces. Fourier Series. The Fourier Transform. Discrete Fourier Analysis. Wavelet Analysis. Multiresolution Analysis. The Daubechies Wavelets. For anyone interested in Wavelets and Fourier Analysis. [via]
Achieving Tabe Success in Mathematics, Tabe 9 and 10 Level D 9780077044695 ISBN: 007704469X Publisher: McGraw-Hill/Contemporary Summary: TheAchieving TABE Successfamily is designed to provide complete skill preparation and practice for TABE 9&10, encompassing Reading, Mathematics and Language, for levels E, M, D and A. This series of books will help students achieve NRA gain through targeted instruction that specifically addresses TABE 9&10 skills. Achieving TABE Success ...workbookscontain the following features: TABE 9&10 Correlation Charts Each tex...t contains a TABE 9&10 Correlation Chart that links each question to targeted skill lessons, enabling instructors and students to build a personalized study plan based on skill level strengths and weaknesses. Pre-tests and Post-tests Each workbook begins with a pre-test and a skills correlation chart to help diagnose strengths and weaknesses and determine TABE readiness. The format of each pre- and post-test matches that of the actual TABE test. Targeted TABE Skill Lessons Each lesson specifically targets a TABE skill. Students work with the innovative lesson format that provides step-by-step instruction to help insure success. The Mathematics lessons offer plenty of instruction and practice to help master each TABE skill. In the Reading and Language workbooks, the lessons are divided into four parts for a graduated approach to learning: Introduceclearly defines, explains, and illustrates the skill, and includes examples. Practicepresents work related to the skill just introduced. Applyreinforces the skill through activities and exercises. Check Upevaluates student comprehension. Unit Reviews and Math Glossary Unit reviews are divided into two parts: Review, which summarizes unit content, and Assessment, to determine student understanding. Mathematics texts contain a Glossary of Common Terms to help students with the language of math. Achieving TABE Success in Reading ...Readersare coordinated with their respective Reading workbooks, to strengthen skills by applying examples and questions that are pertinent to the skill covered in the workbook. Text/TABE Level Content Level Level E 2.0 - 3.9 Level M 4.0 - 5.9 Level D 6.0 - 8.9 Level A 9.0 - 12.9 Contemporary is the author of Achieving Tabe Success in Mathematics, Tabe 9 and 10 Level D, published under ISBN 9780077044695 and 007704469X. Two hundred twenty three Achieving Tabe Success in Mathematics, Tabe 9 and 10 Level D textbooks are available for sale on ValoreBooks.com, one hundred twelve used from the cheapest price of $9.23, or buy new starting at $25
Basic College Math - With Workbook (Custom) - 8th edition Summary: With its complete, interactive, objective-based approach, Basic College Mathematics is the best-seller in this market. The Eighth Edition provides mathematically sound and comprehensive coverage of the topics considered essential in a basic college math course. Furthermore, the Instructor's Annotated Edition features a comprehensive selection of instructor support material. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students int...show moreeract with and master the concepts as they are presented. This approach is especially important in the context of rapidly growing distance-learning and self-paced laboratory situations. * Study Tips margin notes provide point-of-use advice and refer students back to the AIM for Success preface for support where appropriate. * Integrating Technology margin notes provide suggestions for using a calculator in certain situations. For added support and quick reference, a scientific calculator screen is displayed on the inside back cover of the text. * Aufmann Interactive Method (AIM) Every section objective contains one or more sets of matched-pair examples that encourage students to interact with the text. The first example in each set is completely worked out; the second example, called 'You Try It,' is for the student to work. By solving the You Try It, students practice concepts as they are presented in the text. Complete worked-out solutions to these examples in an appendix enable students to check their solutions and obtain immediate reinforcement of the concept. While similar texts offer only final answers to examples, the Aufmann texts' completesolutions help students identify their mistakes and prevent frustration. * Integrated learning system organized by objectives. Each chapter begins with a list of learning objectives that form the framework for a complete learning system. The objectives are woven throughout the text (in Exercises, Chapter Tests, and Cumulative Reviews) and also connect the text with the print and multimedia ancillaries. This results in a seamless, easy-to-navigate learning system. * AIM for Success Student Preface explains what is required of a student to be successful and demonstrates how the features in the text foster student success. AIM for Success can be used as a lesson on the first day of class or as a project for students to complete. The Instructor's Resource Manual offers suggestions for teaching this lesson. Study Tip margin notes throughout the text also refer students back to the Student Preface for advice. * Prep Tests at the beginning of each chapter help students prepare for the upcoming material by testing them on prerequisite material learned in preceding chapters. The answers to these questions can be found in the Answer Appendix, along with a reference (except for chapter 1) to the objective from which the question was taken, which encourages students who miss a question to review the objective. * Extensive use of applications that use real source data shows students the value of mathematics as a real-life tool. * Focus on Problem Solving section at the end of each chapter introduces students to various problem-solving strategies. Students are encouraged to write their own strategies and draw diagrams in order to find solutions. These strategies are integrated throughout the text. Several open-ended problems are included, resulting in more than one right answer and strengthening problem-solving skills. * Unique Verbal/Mathematical connection is achieved by simultaneously introducing a verbal phrase with a mathematical operation. Exercises following the presentation of a new operation require that students make a connection between a phrase and a mathematical process. * Projects and Group Activities at the end of each chapter offer ideas for cooperative learning. Ideal as extra-credit assignments, these projects cover various aspects of mathematics, including the use of calculators, collecting data from the Internet, data analysis, and extended applications. * Eduspace helps instructors take the proven Aufmann Interactive Method to the next level. Eduspace provides instructors with online courses and content in multiple disciplines. By pairing the widely recognized tools of Blackboard with high-quality, text-specific content from Houghton Mifflin, Eduspace makes it easy for instructors to create all or part of a course online. Homework exercises, quizzes, tests, tutorials, and supplemental study materials all come ready to use. Instructors can choose to use the content as is, modify it, or even add their own. Students using Eduspace can review and reinforce concepts with interactive tutorials, prepare for tests using practice exercises, and access all material 24 hours a day. * The Instructor's Annotated Edition features a reduced version of the student text with point-of-use instructor resources in the margins. These include Instructor Notes, In-Class Examples, Concept Checks, Discuss the Concepts, Optional Student Activities, Quick Quizzes, Answers to Writing Exercises, and Suggested Assignments, as well as lists of new or review Vocabulary/Symbols/Formulas/Rules/Properties/Equations. Answers to all exercises are also provided59 +$3.99 s/h Good HippoBooks-DB Toledo, OH No comments from the seller $18.39 +$3.99 s/h Good HawkingBooks Lake Arrowhead, CA 0618586806 Meets the good condition guidelines. Has a large tear. Has wear. Five star seller - Ships Quickly - Buy with confidence not
Book summary Provides the essential mathematical tools and techniques used to solve problems in physics. This work includes differential forms and the elegant forms of Maxwell's equations, and a chapter on probability and statistics. It also illustrates and proves mathematical relations. [via]
Description of Math in Focus 2nd Semester Kit by Saxon/Harcourt Saxon's Math in Focus program is the American version of the highly popular and successful Singapore Approach to mathematics. This approach places strong emphasis on problem-solving and model-drawing skills, with definitions of core math concepts explained in extremely simple terms. Instructional advantages to the Math in Focus program include paced instruction for optimum mastery of concepts, consistent use of visual models and manipulatives for concrete/abstract awareness, and problem-solving methods that help students visualize and understand math concepts
Elementary Geometry 9780471510024 ISBN: 0471510025 Edition: 3 Publisher: John Wiley & Sons Inc Summary: Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic ...geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry. Gustafson, R. David is the author of Elementary Geometry, published under ISBN 9780471510024 and 0471510025. Three hundred sixty four Elementary Geometry textbooks are available for sale on ValoreBooks.com, one hundred three used from the cheapest price of $94.79, or buy new starting at $161.94.[read more] Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). Has some shelf wear, highlighting, underlining and/or writing. Great used condition. We are a tes... [more]RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks).
Ace The CXC/CSEC Mathematics Exam This Year. This eCourse Will Show You How To Answer Questions To Get Full Marks! Course Overview: This eCourse is a digital product built for CXC/CSEC Mathematics Exam Preparation. Inside you'll learn how to answer exam style questions from each of the 11 important topics that you are required to know. The course comprises of step-by-step video solutions and detailed solution notes that walk you through example questions and teach you how to get full marks. Auto-correcting CXC style practise quizzes are also available for you to attempt. Auto-Correcting Practise Quizzes Screenshots: Course Details: Meet Jordan Hello, my name is Jordan Khan and I'm going to be your tutor throughout this eCourse. I'm glad that you've discovered this website and are taking the time to learn more about the course, it shows that you are interested in the program. Our team has worked extremely hard to create this program for you, however it is up to you to take advantage of this opportunity and complete the eCourse with diligence. If you commit to this 5 week eCourse and follow our guidance you will certainly succeed in your exams this year. Even if you only register for the free version of the eCourse, you are still on the right track and hopefully you will learn a lot. I'll be with you every step of the way if you need extra help and I'm happy to answer any questions you might have. So please feel free to leave comments or ask questions if you ever get stuck. Did you know that 70% of all students failed The CXC/CSEC Mathematics exam last year? Over the last 5 years the Mathematics pass rates' across the region have continued to drop and more students are failing the subject. This motivated us to create the Ultimate 5 Week Exam Training eCourse to help you get prepared and learn how to pass the exam! The Ultimate Exam Training eCourse is divided into 11 Topics, each one of these topics cover a collection of CXC/CSEC Style questions and includes full step-by-step video solutions, study notes and auto-correcting exam style practise quizzes for you to test yourself. Week 1: Computation/ Consumer Arithmetic Algebra I Set Theory & Venn Diagrams Week 2: Measurement & Construction Statistics Trigonometry I Week 3: Geometry (Transformations) Number Theory/ Algebra II Algebra, Relations, Functions & Graphs Week 4: Geometry II Trigonometry II Vectors & Matrices Week 5: Revision The Exam training eCourse is an automated series of lessons that teach you how to answer CXC/CSEC Mathematics exam style questions through step-by-step video tutorials and detailed study notes. The course covers the 11 important topics that will appear on the exam paper. When you register for this eCourse you will begin by learning the content available for "Week 1″, which are the first 3 topics. Every week after that, you will continue to automatically unlock more topics until you have unlocked and completed them all. This automated system was put in place to ensure you progress at a steady pace and it allows you to focus on specific subject areas without being overwhelmed with too much all at once. *For the best results we suggest you log on 5 times a week and complete 2 – 3 lessons each time. This should take you roughly 1 hour per day. Contact Us 1 (758) 284 3552 (10am – 6pm, Mon – Fri) How Can We Help? Solve the equation before clicking "Send"5 + 2 = 100% Money Back Policy We try our best to provide quality material and service but if you are not satisfied for any reason, let us know within the first 3 weeks and we'll refund you. Safe & Secure Payment Processing By 2CheckOut 2Checkout is PCI Level 1 certified (the highest level certification possible). So you can trust that your credit card information is safe. The registration deadline is April 15th 2014. (Registration may close before this date if we reach full capacity) Join Us Today! SAVE 20% Right Now! Recently Active Members Student Reviews: Bishop Anstey High School, Trinidad: "This website is amazing and helps because there isn't another one like it that caters to CXC math." Munro College, Jamaica: "The explanation and demonstration were very helpful. I used the videos to explain one of the topics that I had found difficult in class. Two thumbs up!!"
My new plan in introductory physics is to require students to demonstrate some ability to create a numerical calculation. Just to be clear, by "numerical calculation" – I mean to solve a problem by breaking that problem into many simpler problems. Typically, these problems are broken into small time steps and a computer is used to do all the boring calculations. Previously, I have emphasized the idea of numerical calculations in class and even encouraged students to create their own. Starting now, that plan has changed. For me, it is no longer an optional topic. It is right in there along with the Momentum Principle and the Work-Energy Principle. Well, it isn't really a principle, it's more like a tool. I think that leaving numerical calculations out of an introductory physics course is like leaving calculus out of the same course. Sure you can do it, but it's better if you include it. In the past, there have been larger barriers for students to get into a numerical calculation such that many faculty just say "oh well". This is no longer true (in my humble opinion). There are super simple tools like VPython and GlowScript (both excellent options). So there is no more excuse. And yes, I am talking to both students and faculty. Numerical calculations are just part of the way things are done. It would be a shame for students to never get a chance to practice this method. Create a numerical calculation to solve some problem. You can use VPython, or GlowScript or really anything. I give several examples and a tutorial using VPython. The student will get 3 out of 5 points if they create and show a program that models the motion of an object with a constant force (like a fan cart on a track or a projectile motion problem). The student will get 4 points for a numerical calculation that involves a non-constant force (like the Earth orbiting the Sun). Finally, the students get a maximum score of 5 out of 5 if their numerical calculation is for some type of problem that can not be easily solved analytically (like the 3 body problem or projectile motion with air resistance). Once the students turn in the assessment by the due date, they can continue to improve their code over the semester to increase their score. That's it. Of course, there are still some problems. Some students wait until the last minute and then have some type of technical difficulty. Other students spend time to create a program, but it ends up just being some type of input-output calculator instead of a numerical calculation. But overall, it's great fun. Students come in with interesting ideas and the discussions almost always revolve around physics ideas instead of coding and syntax errors. But WAIT! I can't do this because I'm not a computer scientist. This is wrong. This numerical calculation is not computer science. When I think of computer science, I think of creating and exploring new algorithms in computer code. Creating a numerical calculation is just coding. Computer science is much more than just making code. It's also wrong another level. Student's don't have to be experts at computer coding to write a program. In fact, the great thing about humans is that they don't JUST do ONE thing. Instead, they do all sorts of things. Here are some other comments about physics classes. You don't have to be an artist to draw a free body diagram. You don't have to be an author to write an essay that explains your physics problem. You don't have to be a mathematician to evaluate an integral of velocity to find the change in position. You don't have to be a public speaker to share your physics solution with the rest of the class. You don't have to be a physicist to like physics. I can just imagine the first time a physics faculty told a class that they needed to draw a free body diagram of the forces on an object for the physics solutions. I wonder if a student complained that this was supposed to be a physics class and not an art class. Nature of Science and Numerical Calculations Let me just add a couple of ideas about this. What is science all about? It's all about building models. Well, it's not only about building models. In order to call it "science" we have to see if that model agrees with real life. This checking for agreement is called an experiment. I think most people will agree with my above definition of science. If so, you have to ask yourself "where do I put numerical calculations"? Are they experiments? Are they real life? Are they models? Well, they sure as heck aren't real life (except for Flappy Birds – that's real). If a numerical calculation doesn't agree with real life, we like to call it something else. Suitable names would be either "art" or "computer game". If the calculation does agree with real life, we call it a model. In the end, students are mostly happy Yes, many students had problems along the way to create their first numerical calculation. They started off confused and intimidated. But for those that pushed through these problems, they came out with their own creation. They had a program that ran and that they were proud of. It's great fun. Shani Davis of the U.S. skates in the prototype of the official US Speedskating suit. Image: Pavel Golovkin/AP It seems the USA speed skaters aren't doing as well as they had hoped to in the 2014 Winter Olympics. One idea is that the new uniforms could be part of the problem. I don't know much about skating, so I'm just going to do a rough estimation of the effect a uniform could have on speed skating. Remember, I said I don't know much about skating and this is just an estimate – you know, for fun. Constant Power Humans Let me start with a simple model of a speed skater traveling at a constant speed. In this model, I will assume the following: During a race, a human skater has a constant power output of some value P. There is an air resistance force on the skater that is proportional to the velocity squared. The air resistance force is also proportional to the density of the air and some drag coefficient. The drag coefficient depends on both the shape of the skater and the surface properties of the suit. What does the human use all this power for? Two things. First, is moving arms and legs and stuff like that. The second is to work against the air resistance. I will call the fraction of power that goes towards the air resistance f. Let's say that I have a skater with a power fP (so this is just the power fighting the air drag – but I will just call it P from now on) moving at a constant speed of v and all of this power is going into fighting the air resistance (which totally isn't true – but this should still work out ok). Now let's say the skater moves a distance s over some time interval. I can calculate the work and the power of the air resistance force and set that equal to the power of the human skater. Now for some values. Let's look at the Men's 1500 meter race. The Olympic Gold medal went to Zbigniew Brodka from Poland with a time of 1:45.006 or 105.006 seconds. This would give him an average speed of 14.285 m/s. Now I just need an estimate for the human power. In this previous post on bikes going up hills, I used a power of 300 watts. I am going to bump that up to 350 watts because – you know, it's the Olympics and these guys are giving it EVERYTHING THEY GOT. If I just totally guess a air drag fraction of 0.25, this would put the air drag power at 87.5 Watts. With this, I can get an estimate for all the non-velocity stuff in the previous equation (the product of the density, cross sectional area, and drag coefficient). Maybe I could just call all of this stuff "K" (oh, might as well throw the 1/2 in there too). I can solve for this K value for the top Olympic performer and I get 0.030017 kg/m. Ok, here is the fun part. Suppose that I increase this air drag stuff by just a super small amount. What effect would that have on the total time? Again, a reminder that I am making some assumptions here that aren't absolutely valid but will still give me an estimate of the effect of these things. Here is a plot of the race times for a human with constant power but an increase in drag stuff. I have included points with the times of top 10 skaters (with calculated a values). What does this say? It says that if all of these skaters were identical except for air drag, there would be about a 3 percent increase in drag from the first place skater to the 10th place. From this estimation, drag could be a significant factor in the race. The key words here are "could be". This is just an estimate that ignores many things. Constant Drag Coefficient Humans What if I pretend like all the humans have the same body shape and uniforms so that they have the same drag coefficient? What if the only difference in race times is due to differences in human power output? Here is a plot of the race time as a function of a percent of the 87.5 watt power that fights the air. From this function, it seems like a human with power of around 85.14 watts instead of 87.5 would be able to account for the a 1 second longer race. What does this mean? This means that differences in human power could also easily account for differences in time. Is this crazy to even approach a problem this way? No. These assumptions are a type of spherical cow (here is the spherical cow reference for non-physicists). Even if the assumptions are totally crazy, they at least give us a place to start. From this, it seems reasonable to further explore the role of air drag in speed skating (which surely someone has already done). What would be the next step? I would get a speed skater and start making changes to things (like power and uniform) and see what kind of times that produces. Yes, it isn't so simple to change things when dealing with humans, but it's a human sport. twitter. This lead to the following reply from Peter: CHALLENGE ACCEPTED. So, what physics could I use to say that Peter's hoodie changes from the color red to pink? What is Color? Maybe I should change that title to "how do we see color?" Every thing that we see is a result of two possible things. Objects can emit light (like a lightbulb) or objects can reflect light (like an apple). You are probably reading this blog post on a computer screen. In that case, there is most likely a light behind your display that emits light. That light enters your eye and your brain interprets the different wavelengths of light as color. If you had this post printed out on actual paper, you would see it because light reflects off the paper and then enters your eye. For this second case, if you turn off all the surrounding lights there would be no light hitting the paper. This means that no light reflects off the paper and you couldn't see it. It doesn't matter if an object emitted the light or the light reflected off the object, the color your brain interprets is based on the wavelength of the light. Of course, our human eyes can only see a small range of light that we call the visible spectrum. You can see this spectrum on your own by taking one of these inexpensive prism glasses. When light passes through them, different colors bend different amounts so that you can see all the colors. Here is an image using my phone's camera to look through one of these special glasses. Image: Rhett Allain If you think you see pink in this spectrum, you could be wrong (but I think I see something pinkish in there). However, this image was created with a camera so that sometimes you don't always get what is actually there (maybe). The point is that there a range of colors that correspond to different wavelengths and pink isn't one of them. How do we see the color pink? If you take all of these colors of light together, your brains says that it is the color white. If you just take the red and the blue, your brain says "PINK". The same is true for other colors like "brown" and "olive". They are all combinations of wavelengths of light. What about the Doppler Effect? This was my first idea of changing the color of a hoodie. It turns out that when an object moves towards or away from an observer at very high speeds, the apparent wavelength of light shifts. An object moving AWAY from an observer makes the wavelength of light longer than if it were a stationary source. Since red light has a longer wavelength than other colors, we call this effect a red-shift. If an object was moving towards you at a very high speed, the opposite would happen. The wavelengths of light would be shorter and make it appear more blue. Here is my attempt at showing a green object that is both stationary and moving. So that is the Doppler effect with light. It's a real thing and it is used in astronomy to measure the speed that stars are moving away from us. It can also be used to measure the wobbling motion of a star caused by a planet orbiting it. At first glance it seems that the Doppler effect couldn't be used to change Peter's hoodie from red to pink. Sure, he could be moving away from us at really high speeds, but that would shift the color of his hoodie to longer than red wavelengths (infra red). If he was moving towards us, it could appear different colors, but it would have to be one of the colors of visible light. He could be moving very fast towards us such that the he appeared green or blue, but not pink since you can't shift the red wavelengths to something that is a combination of wavelengths. Really, the only way I can think to make the Doppler shift turn Peter's hoodie to the color pink is if part of him is stationary and part of him is moving away from us really really fast. The part that moves away would be blue shifted all the way to a blue wavelength. The rest of Peter would still have a stationary and red hoodie. For the observer, there would be a combination of red and blue wavelengths making the hoodie appear pink. However, there is one problem. Breaking up Peter into parts is a bad idea. If you would like another view of the color pink, check out this interesting video from Minute Physics. But maybe Peter just took off his red hoodie and put on the pink one. I know that's a boring explanation, but it's one we should consider. Just in case you want to read some more about light, here are some related links for you. In the Olympic event of Short Track Speed Skating, athletes race around a rather short track (thus the name of the event) that only has a circumference of 111 meters. It always seems almost magical the way the skaters lean over so far while going around the turns. The faster the skater goes around the turn, the more he or she leans. Clearly, the skater can't lean over 90°, right? So what is the maximum speed? Here are some of the basic physics ideas: Since the skater is turning in a circle, there must be a force pushing on the skater in the direction of the center of the circle. This force is the frictional force on the side of the skate. If a skater didn't lean at all, this frictional force at the feet would cause the skater to tip over. By leaning, the skater can balance the torque from gravity with the frictional force. This seems like it would work, but I think you have to also consider the acceleration while turning. I suspect that at greater lean angles, there is less frictional force on the skate. There is one other big idea to use here. Fake forces. Yes, they can be dangerous and many introductory courses explicitly tell you they are bad – but they can be useful. What is a fake force? Well, the normal ideas of force and motion say that forces change the momentum of an object. But these rules only work if the reference frame is not accelerating (called an inertial reference frame). If you are moving along with a turning skater, you are in an accelerating reference frame. You can still make the force rules work, but you have to add a fake force. You are already familiar with fake forces. You use them all the time. When you are sitting in your car and push the gas pedal, the car accelerates forward. Since you are inside the car, you are in a non-inertial frame. So, what pushes you back into your seat as you accelerate? The answer: nothing. You can pretend like there is a force that is equal in magnitude to the force of the seat pushing you forward, but this is a fake force. In general, you can write this fake force as: Ok, back to the turning skater. Here is a force diagram including the fake force. So, just 4 forces. For an object in equilibrium (which would be true in our non-inertial reference frame), the following must be true: The net force in both the x- and y-directions must be zero as well as the torque (which we can treat as a scalar here) about some point. Here, I will use the scalar definition of torque about some point o: Where F is the applied force, r is the distance from the force to the point "o" and θ is the angle between F and r. If you want to know more about torque, this older post might be useful. Oh, one more thing. What about the fake force? This depends on both the mass of the skater and the acceleration of the frame (which is also the skater). Since the skater is moving in a circular motion, the magnitude of this fake force would be: Here v is the magnitude of the skater's velocity and R is the radius of the circle the skater is moving in. Now, there is a trick here. If the skater is leaning in towards the center of the circle, different part's of the body will be at different distances from the center. If the radius is large enough, these distance differences won't really matter. For the rest of the calculations, I will assume this fake force acts at the center of mass of the person. Now I can start to put in some values. If we look at the total torque about the point where the skates touch the ice, I can ignore both the normal and frictional forces since they produce no torque. Oh, let me assume that the center of mass is in the middle of a skater with a height of h. This gives: This says a few things: The faster you go, the more the skater leans (smaller angle). Skater mass doesn't matter. The height of the skater doesn't mater either. How about a plot? We need a value for the radius of the circle. It seems like this is around 8 to 8.5 meters (depending on where the skater turns). I will go with a value of 8 meters. Here is the lean angle as a function of speed. I suspect you really can't get much lower than a 20° angle when leaning. That would put the maximum speed at about 14.7 m/s (or 32.9 mph). That's probably faster than the skaters normally go – but let me just check. The world record for the 500 meter track is a time of 39.937 seconds. This would give an average speed (assuming the skater actually went 500 m – which probably isn't true) of 12.5 m/s. Based on this value, I am going to put the best lean angle at 35° for a turning speed of about 10.6 m/s (and then the skaters could go faster on the straight away). But what about friction? In this model, the friction force doesn't change as the skater leans. If I assume the typical friction model that says the frictional force is proportional to the normal force, then we need to look at hte normal force. There are only two forces in the vertical direction: the normal force and the gravitational force. These must add up to zero and don't depend on the lean angle. So based on this, the maximum frictional force is just some value. We can use this to find the coefficient of friction between the sideways pushing skate blade and the ice since the friction force must be the same magnitude as the fake force. Note that this model of friction probably doesn't work for a blade cutting into ice. With a radius of 8.0 meters and a speed of 14.7 m/s, I get a coefficient of static friction with a value of 2.76. Typical values of the coefficient are usually between 0 and 1. So, this seems a little crazy. However, there is another way to get a value for the coefficient of friction between the skate blade an ice (for perpendicular motion). When the skaters start from rest, they use the side of the blade to push on the ice. This frictional force is what causes them to increase in speed. By measuring the acceleration, you could get another estimate for the coefficient of friction. I will save that for either another post or you can do it for homework. Hint: use video analysis. In some news item, it was pointed out that an Olympic gold medal is made with only 1% gold. Ok, that makes sense. A full 100% gold medal would probably be quite expensive. Also, gold is a rather soft metal so it might get bent or scratched too easily. If Not Gold, then What? Now for the fun part. Let's say that the medal is 1% gold (by mass). What is the density of the rest of the stuff? To start, I will write the total mass of the medal as: Hopefully it is clear that the m subscript represents the medal. I already know the mass of this other stuff in the medal, but I need its volume. If I subtract the volume of the gold from the total volume, that will just be the other stuff. Now I can solve for the density of the stuff. That's it. It's all finished except for the numbers. If I put the values from above into this expression, I get a stuff density of 6.717 g/cm3. Based on this list of metal densities, it seems like that stuff could be several things. It's closest to the density of Cerium – but Wikipedia lists it as being both soft and it oxidizes in air. It could be cast iron or zinc. Modern pennies have a zinc interior, so maybe the medal is just like a huge penny. Cost of a Gold Medal The gold medal has a mass of 531 grams. If it was 100% gold, it would have a mass of 1.52 kilograms. Damn. That's 3.35 pounds. You would surely feel that sucker around your neck. Wouldn't you? Next question: why don't they make Olympic Gold Medals out of gold? I think the answer is obvious. A Historical Look at Medals Sometimes I can't stop. Using the data from that Wikipedia page, I can plot the medal volume as a function of year (for both Winter and Summer Olympics). Why did I add a linear fit? I guess just because I could. However, you can use it to answer the following questions: If both the Winter and Summer Olympic medals continue this trend, how big will the medals be around the year 2050? Could you get some type of exponential model to fit this data better than a linear fit? Why do the medals seem to get bigger over time? Is bigger better? I wonder if there is any correlation between medal size and location of the Olympics. Go ahead and play with it. Plotly lets you try your own fits and make this into your own graph. That's what makes Plotly so awesome. Oh, I left off a couple of medals that weren't cylindrically shaped. What about the medal density. There doesn't seem to be a trend for the density so I will just plot it as a histogram. Look at the medal for Summer Olympics in 1912 with a density of 18.26 g/cm3. That looks close to pure gold. Or maybe it's mostly lead. One more homework question. How much would the 2014 medal be worth if it were made of Bitcoins? Yes, I know that Bitcoins aren't really real. That just means that you have to be creative to come up with an answer.
Basic College Mathematics - 4th edition Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success838508 +$3.99 s/h New Campus_Bookstore Fayetteville, AR New Annotated Instructor's Edition. Book has the same contents as the student edition, but includes answers. 4th Edition Ships same or next day. Expedited shipping takes 2-3 business days; standard sh...show moreipping takes 4-14 business days. ...show less $153
OpenAlgebra.com is a free online algebra study guide and problem solver designed to supplement any algebra course. There are... see more OpenAlgebra.com is a free online algebra study guide and problem solver designed to supplement any algebra course. There are hundreds of solved problems, video solutions, sample test questions, worksheets, and interactives. self-paced learning and reference tool will help students understand how to format research papers, journal articles,... see more This self-paced learning and reference tool will help students understand how to format research papers, journal articles, and lab reports; what standards of writing style and ethical research to follow; how to make correct citations when quoting or paraphrasing sources; and how to make a References page, following the documentation style of the 6th Edition of the APA Manual.
Houghton Mifflin Harcourt's Math in Focus®: Singapore Math by Marshall Cavendish is the U.S. version of Singapore's most popular and proven math curriculum. Based on the framework developed by the Singapore Ministry of Education, Math in Focus represents authentic Singapore math, draws on best practices from around the world, and highlights problem solving as the focus of mathematical learning. Resulted in exceptional math achievement by Singapore students on international studies Supports the big ideas in the Common Core through a focus on classroom learning, discussion, practice, and problem solving Offers an effective, research-based pedagogy and balances conceptual understanding, visual learning, and problem solving to help all students achieve greater success. Book Description:Houghton Mifflin Harcourt Publishing Company. Book Condition: Brand New. Bookseller Inventory # 366727
... read more Abstract Algebra by W. E. Deskins Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problemsIntroduction to the Theory of Sets by Joseph Breuer, Howard F. Fehr This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 editionGroup Theory: The Application to Quantum Mechanics by Paul H. E. Meijer, Edmond Bauer Upper-level undergraduate and graduate students receive an introduction to problem-solving by means of eigenfunction transformation properties with this text, which focuses on eigenvalue problems in which differential equations or boundaries are unaffected by certain rotations or translations. 1965 edition. Problems in Group Theory by John D. Dixon Features 431 problems in group theory involving subgroups, permutation groups, automorphisms and finitely generated Abelian groups, normal series, commutators and derived series, solvable and nilpotent groups, and more. Full solutions. 1967 of rings, and polynomial rings. 1969
Essential Calculus: Early Transcendentals Book Description: This book is for instructors who think that most calculus textbooks are too long. In writing the book, James Stewart asked himself: What is essential for a three-semester calculus course for scientists and engineers? ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS, Second Edition, offers a concise approach to teaching calculus that focuses on major concepts, and supports those concepts with precise definitions, patient explanations, and carefully graded problems. The book is only 900 pages--two-thirds the size of Stewart's other calculus texts, and yet it contains almost all of the same topics. The author achieved this relative brevity primarily by condensing the exposition and by putting some of the features on the book's website, Despite the more compact size, the book has a modern flavor, covering technology and incorporating material to promote conceptual understanding, though not as prominently as in Stewart's other books. ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS features the same attention to detail, eye for innovation, and meticulous accuracy that have made Stewart's textbooks the best-selling calculus texts in the world
Short Description for A Student's Guide to Vectors and Tensors Explains vectors and tensors in plain language to give undergraduate and beginning graduate students a better understanding. Full description Full description for A Student's Guide to Vectors and Tensors Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.
NOTE: Water and minor coffee stain and wrinkling to the edge of pages. Otherwise in good and clean reading condition. Not pretty on the edges but OK to read. Eligible for Free Super Saver Shipping. Quick Amazon Shipping plus Hassle Free Returns. Your 100% Satisfaction is Guaranteed! {"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":93.11,"ASIN":"0395977223","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":72.7,"ASIN":"0395977274","isPreorder":0}],"shippingId":"0395977223::aTYnpFyNs75AfR8kmQqu01Ygk2HjrptBxdPGWKURFMI9Az32AUlc42nxrs4xCrnnevRiTYaFMqonbJPUskbE8TGcO8IsC7U2jf5z93SzXvo%3D,0395977274::L2%2BCRBuGaLh1ZDk2th6K43RkMrKwALFYRun8paKHAUViR4wYs0FPVzZarDMp5W07M70RMn5rt63tpKeAytpwwIlwUT%2BkjtT8%2B48kSM%2Bwo0 well thought out Algebra textbook. I started teaching from it earlier this year and have grown to appreciate it very much. I am especially pleased the more I compare this to other books. Most math books these days are little more than pretty colors, crazy fonts, neat pictures, lots of distractions, and very little actual math instruction. This book, on the other hand, actually focuses on the math, explains it reasonably well, and has great sets of real example problems and practice problems of all difficulty levels, including lots of practical problems that aren't forced or contrived. It is not perfect, but it is the best I have seen. Pros: Lots of math rather than lots of silly distractions Appropriate difficulty level - not "dumbed down" Good examples worked out and explained Good problem sets for homework and practice Cons: Some explanations are just a bit short, although most are good Factoring, the hardest topic, is early in the book, which makes it difficult for some students This is one of only a handful of books that actually covers Algebra 1 in detail, and in an order that builds a strong foundation along the way. Sure, there will be students who find this book hard to understand -- they are the ones who are not really ready for algebra and should be either taking pre-algebra or one of the many "algebra lite" books that are available. Well-prepared students will find this book a great resource, and should find themselves extremely well-prepared for future classes in mathematics. It is arguably one of the best algebra books available, and is one of the top algebra textbooks as rated by Mathematically Correct, a group that is concerned with the quality of mathematics instruction in California (and beyond). This is a pretty good comprehensive textbook for students of all levels. There are twelve chapters in the book, plus a glossary, various helpful appendices, and an index. The book contains "A" (easy), "B" (intermediate), and "C" (challenging( problem sets for every topic covered. Also, enrichment topics are included, including "Extra," "Challenge," "Historical Note," "Application," and "Career." Like any other book, this book does, however, isn't absolutely perfect. One thing about this book is that the explanations in this book at the top of every section summarizes important concepts and gives some examples and definitions; however, the examples often only cover the "A" and half of the "B" problem sets, and the "C" problems are often unexplained. Also, the textbook seems to be designed for slightly more advanced students, as the problems and pace of the text isn't for beginning or struggling students. One note is that this book isn't the best tool to teach yourself math, although it is an excellent source for problem sets, and also serves as an excellent reference for algebraic concepts. I would recommend "CliffsStudy Solver Algebra I" if you are having trouble with Algebra I or if you want to teach yourself math, because this textbook is not designed to be a tutorial. I have tried to teach myself many new math concepts, but this book didn't always help - though I did some new problem solving strategies from here. Sometimes the explanations were simply skimmed over too quickly. Overall, this is a great book. It's really helped me with my math grade, but this is just one of my math books which helped me. With teacher guidance, one or two more study guides, and most importantly, a dedicated self-motivated mind, you are bound to be a success in algebra. I think overall, the book, Algebra Structure and method is wonderful. The book provides the odd answers to the problems, in the back, which is nice, because it gives me sense of if I'm getting the problems correct, without telling me all the answers. I also like how the difficulty of problems is catagorized, with "a" problems being the easiest, "b"problems, in the middle and "c" problems the hardest. Sometimes the explanations of how to do the problem are a little confusing,though. All in all, this book is a good teacher. This book is a comprehensive algebra 1 course despite the implications of the words "Book 1" in its title. You can use it for self-instruction, because the answers to the odd-numbered problems are given at the back of the book -- and there are lots of problems. It is also a great source of supplementary material for students who are taking algebra 1 in school. Supplementary material is necessary because most algebra 1 courses these days are "dumbed down." The District of Columbia Public Schools, for example, uses a book (ISBN 0618250182) that has been stripped of traditional problems that show students the power of algebra -- "work problems," "mixture problems," "age problems" and so on. And it has been stripped of difficult problems in all of the topics that remain. The book under review here covers the traditional topics and has difficult problems in all topics. The book would be good for a differentiated course, because the problems in each section are grouped into three levels of difficulty, labeled A, B and C.
Winthrop Harbor StatisticsBesides giving you concepts and practice problems, I can also explain the applications of discrete mathematics and why it is such an important branch of mathematics in several technical fields such as computer science and engineering. I have taken several higher level mathematics courses during ... am a University of Illinois Engineering Grad and I always had A's or B's in the core curriculum courses while I was there. I may have had a C or two, but they were always in the mandatory classes that were of no real relevance to my core, such as History or social studies. I still remain in touch with my core and use them in everyday life
157685552X 9781576855522 501 Algebra Questions:Using a self-paced, multiple-choice approach that moves from basic questions to more difficult ones, this easy-to-use book teaches candidates to successfully prepare for all the different types of algebra questions on standardized tests. The book covers the full range of math concepts and properties, including exponents and radicals; algebraic expressions and integers; multiplying and factoring polynomials; and using the quadratic formula to solve equations. Included are plenty of tips on how to avoid careless mistakes and strategies for doing the best on any test. Back to top Rent 501 Algebra Questions 2nd edition today, or search our site for textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by LearningExpress, LLC.
and Complex Analysis This is an advanced text for the one- or two-semester course in analysis, taught primarily to math, science, computer science, and electrical ...Show synopsisThis is an advanced text for the one- or two-semester course in analysis, taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level 0070542341 Brand New Paperback Overseas International...New. 00705423419. Hardcover, McGraw-Hill Description:Very Good + in Very Good + jacket. Hardcover with unclipped...Very Good + in Very Good + jacket. Hardcover with unclipped dust jacket in an old plastic protector, undated [c1970, appears to be a Chinese pirated edition), xi + 412 pages; very light shelf wear, bumps to corners, ink signature, ink and/or pencil markings on approximately 55 pages, but gently used, otherwise very clean and unmarked. Description:Very Good+ Very Good+ hardcover in forest green boards. Normal...Very Good+ Very Good+ hardcover in forest green boards. Normal shelfwear to the exterior including rubbing/creasing to the head and the foot of the spine. General smudging/fading to the exterior. Foxing to the page-ends. Previous owner's name and date, in orange marker, to the head of the front free endpaper; slight crease down the front free endpaper. Otherwise, pages and text are clean and bright. Overall, an attractive copy. 412
Blue Bell Precal understanding of algebra is a foundational skill to virtually all topics in higher-level mathematics, and it is useful in science, statistics, accounting, and numerous other professional and academic areas. 1. Describe basic operations or numbers and signs. 2. Solve simple to complex fraction problems. 3
MAT 521 Geometry for Teachers This is a sample syllabus only. Ask your instructor for the official syllabus for your course. Instructor: Office: Office hours: Phone: Email: Revised Course Description Topics from geometry including: points and lines in a triangle, properties of circles, collinearity, concurrence, transformations, arithmetic and geometric means, isoperimetric theorems, and reflections principle. Moreover, this course is the study of geometry as an axiomatic system, which includes the study of postulates, theorems, formal proofs, rules of congruence, angle measurement, similarity, parallelism, and perpendicularity. Furthermore, topics in the research of how students learn geometry and effective teaching methods of geometry will be studied. MAT 521 meets for three hours of lecture per week. Prerequisites Graduate standing and one year of full time secondary teaching. Objectives After completing MAT 521 the student will appreciate the pervasive use and power of reasoning as a part of mathematics deduce properties of figures using transformations and using coordinates identify congruent and similar figures using transformations analyze properties of Euclidean transformations and relate translations to vectors be knowledgeable of current technologies relevant to geometry instruction be knowledgeable of the current research and theories of geometry instruction. Expected outcomes Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 521. Method of Evaluating Outcomes Evaluations are based on problem solving and reasoning performance tasks, homework, projects, papers, class presentations, short tests, portfolio of total work for the semester, and/or scheduled examinations. Grading Policy Students' grades may be based on homework, projects, papers, class presentations, short tests, and/or scheduled examinations that test students' understanding of the topics covered in the course (see "Method of evaluating outcomes"). The instructor determines the weight of each of these factors in the final grade. Attendance Requirements Attendance policy is set by the instructor. Policy on Due Dates and Make-Up Work Due dates and policy regarding make-up work are set by the instructor. Schedule of Examinations The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes. Academic Integrity The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".) Accomodations for Students with Disabilities Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how I can best help you. All disclosures of disabilities will be kept strictly confidential. Please note: no accommodation may be made until you register with the DSS in WH B250. For information call (310) 243-3660 or to use telecommunications Device for the Deaf, call (310) 243-2028.
Hathorne PrealgebraAlgebra 2 is very important to master, as there are 2 or more years of math in the future. Give your child a much better chance for success in learning these Algebra 2 Topics: 1. Equations and inequalities, 2. ...It is easier for one student to relate to and learn from another student rather than an adult. I
Pre-algebra Topics include a review of whole numbers, fractions, and decimals, a complete development of percent, ratio/proportion, exponents, order of operations, integers, the use of variables, and simple equation solving. Course is graded on a pass/no pass basis. Course does not transfer. Prerequisites: Designated math placement test score; RD30 (may be taken concurrently). A scientific calculator is required. Pre-algebra Reinforces skills in whole number, fractions, and decimals while introducing computation with rational numbers, exponents, order of operation, and the use of variables, expressions, formulas, and equations. Ratio and proportions, percent, and topics in measurement are also studied. Working with real data, formulas, and applications will be stressed. This course is open to students who place into either SK8 or MTH20. It is a one-term experience for these two courses, meeting the outcomes for MTH20 while providing support for motivated students who are place at the SK8 level. It is designed to increase confidence and desire to continue the study of math. Course is graded on a pass/no pass basis. Course does not transfer. Prerequisites: Designated math placement test score; RD30 (may be taken concurrently). Fundamentals of Algebra I Beginning algebra introduces the study and application of real numbers, operations with real numbers, exponents, order of operations with linear expressions, mathematical modeling, solving linear equations, solving inequalities, methods of problem solving, rates, slope, graphs of lines, equations of lines, functions, and systems of linear equations. Working with real data, formulas, and applications will be stressed. Course is graded on a pass/no pass basis. Course does not transfer. Prerequisites: MTH20 and RD30 or designated placement test scores. A scientific calculator is required. There is a significant online component in this class. Applied Technical Math Introduces the study of algebra in an applied work-related context. Course topics include operations with real numbers measurement, ratios proportions, percents, unit analysis, order of operations, solving equations, manipulating formulas, Pythagorean theorem, trigonometry, area, perimeter, surface area and volume. Course is graded on a pass/no pass basis. Course does not transfer. Prerequisites: MTH20 and RD30 or designated placement test scores. A scientific calculator is required, and there is a significant online component in this class. Fundamentals of Algebra II Includes the study and application of exponents, polynomial operations, factoring polynomial expressions, solving polynomial equations, rational expression operations, and solving rational equations. Course is graded A through F. Course does not transfer. Prerequisites: MTH60 and RD30 or designated placement test scores. A scientific calculator is acceptable, but a graphing calculator is recommended. There is a significant online component in this class. Fundamentals of Algebra II Recitation Designed for students needing additional help with MTH65. Course is optional. Graded on a pass/no pass basis. Course does not transfer. Prerequisites: Concurrent enrollment in MTH65. A scientific calculator is required. Intermediate Algebra, Part I Designed for students who need a slower pace for MTH95. Topics include the study and application of compound inequalities, rational exponents, simplifying radical expressions, solving radical equations, solving quadratic equations, and graphing quadratic functions. Course is graded A through F. Satisfactory completion of both MTH93 and MTH94 is equivalent to MTH95. Course does not transfer. Prerequisite: MTH65, Part II Designed for students who need a slower pace for MTH95. Topics include the study and application of graphing quadratic functions, relating exponential and logarithmic functions, solving exponential and logarithmic functions. Course is graded A through F. Satisfactory completion of both MTH93 and MTH94 is equivalent to MTH95. Course does not transfer. Prerequisite: MTH93 Topics include the study and application of compound inequalities, rational exponents, simplifying radical expressions, solving radical equations, solving quadratic equations, graphing quadratic functions, relating exponential and logarithmic functions, and solving exponential and logarithmic functions. Course is graded A through F. Course does not transfer. Prerequisites: MTH65 and RD30 or designated placement test scores. A graphing calculator is required (instructor will be using the TI-83 or TI-84 graphing calculator in class) and there is a significant online component in this class. Intermediate Algebra Recitation Designed for students needing additional help with MTH95. Course is optional. Includes review of MTH65 material, using a graphing calculator, and focuses on topics and concepts of particular difficulty presented in MTH95. Graded on a pass/no pass basis. Course does not transfer. Prerequisite: Concurrent enrollment in MTH95. A graphing calculator is required (instructor will be using the TI-83 or TI-84 graphing calculator in class). Introduction to Contemporary Mathematics Designed for liberal arts students. Includes the study and application of logic and reasoning, problem solving, set theory, geometry, probability, statistics, and math of finance. May also include number theory, systems of equations and inequalities, matrices and determinants, counting theory, and numeration systems. Prerequisite: MTH95. A scientific or graphing calculator is required (instructor will be using the TI-83 or TI-84 graphing calculator in class). There is a significant online component in this class. College Algebra Topics include linear functions and models, quadratic functions and models, graphing polynomials, rational and inverse functions, systems of equations, zeros of polynomials, exponential and logarithmic functions, and conic sectionsCollege Algebra Recitation This is an optional course that can be taken concurrently with MTH111. Provides additional help with MTH111 concepts. Reviews MTH95 material and using the graphing calculator, and covers the topics and concepts of particular difficulty presented in MTH111. Prerequisites: MTH95 or designated placement test score and concurrent enrollment in MTH111. Elementary Functions Includes right triangle trigonometry, trigonometry of the unit circle, solving trigonometric equations, identities, the law of sines and cosines, vectors, parametric and polar equationsElementary Functions Recitation This is an optional course that can be taken concurrently with MTH112. Provides additional help with MTH112 concepts. Reviews MTH95 material and using the graphing calculator, and covers the topics and concepts of particular difficulty presented in MTH112. Graded on a pass/no pass basis. Prerequisites: MTH95 or designated placement test score and concurrent enrollment in MTH112. Special Studies in Mathematics Fundamentals of Elementary Math I w/Lab Designed for students studying to be elementary/middle school teachers and is focused on creating a deep understanding and positive attitude toward mathematics. Topics include problem solving, set theory, number systems, whole number operations, mental math, and number theory. Course is graded A through F. Prerequisites: MTH95 or designated placement test score. A scientific calculator is required. Fundamentals of Elementary Math II w/Lab Designed for students studying to be elementary/middle school teachers and is focused on creating a deep understanding and positive attitude toward mathematics. Topics include fraction operations, decimal operations, ratios, proportions, percents, integer operations, statistics and probability. Course is graded A through F. Prerequisites: MTH95 or designated placement test score and MTH211. A scientific calculator is required. Fundamentals of Elementary Math III w/Lab Designed for students studying to be elementary/middle school teachers and is focused on creating a deep understanding and positive attitude toward mathematics. Topics include two- and three-dimensional shapes, measurement, volume, surface area, congruence, similarity, coordinate geometry, and transformational geometry. Course is graded A through F. Prerequisites: MTH95 or designated placement test score. A scientific calculator is required. Probability and Statistics w/Lab Descriptive statistics covering the nature and presentation of data, measures of central tendency, probability and probability distributions (normal and binomial), confidence intervals, sample sizes, and hypothesis testing. Course is graded A through F. Prerequisites: MTH95 and RD30 or designated placement test scores; a graphing calculator is required (instructor will be using the TI-83 or TI-84 graphing calculator in class). There is a significant online component in this class. Inferential Statistics Builds on the basic knowledge and skills learned in MTH243 and utilizes spreadsheet skills gained in CS125ss. Students will use Excel extensively to solve statistical problem. Emphasis is on the understanding and application of hypothesis testing, analysis of variance (ANOVA), correlation and regression, and Chi-square techniques. Designed to provide students with analytical skills they will need in upper division business courses including accounting, finance, operations management and applied research. Course is graded A through F. Dual numbered as BA282. Prerequisites: BA131, MTH243, and RD30; CS125ss recommended. Calculus I (Differential) w/Lab Topics include limits, the derivative, and applications. Course is graded A through F. Prerequisites: MTH111 and MTH112 or designated placement test scores. A computer lab is required. A graphing calculator is also required (the TI-83, TI-84, TI-89 or TI-92 graphing calculators are recommended) There is a significant online component in this class. Calculus II (Integral) w/Lab Topics include techniques of integration and applications and transcendental functions. Course is graded A through F. Prerequisites: MTH251Calculus III w/Lab Topics include infinite series, polar coordinates, conics, parametric equations, and introduction to vectors. Course is graded A through F. Prerequisites: MTH252Vector Calculus w/Lab Topics include integration and differentiation of multivariable functions and vector calculus. Course is graded A through F. Prerequisites: MTH253Differential Equations w/Lab First course in ordinary differential equations for science, mathematics, and engineering students. Includes first order differential equations, linear second order differential equations, and higher order linear differential equa-tions, with applications. Additional topics include Laplace transforms, series solutions of linear differential equa-tions, and systems of differential equations, with applications. A computer lab is required. Prerequisite: MTH253 or instructor approval. A graphing calculator is also required (the TI-83, TI-84, TI-89 or TI-92 graphing calculators are recommended). Linear Algebra w/Lab Topics include line vectors, n-tuples, algebra of matrices, vector spaces, and linear transformations. Offered on demand only. Course is graded A through F. Prerequisite: MTH252. A computer lab is required. A graphing calculator is also required (the TI-83, TI-84, TI-89 or TI-92 graphing calculators are recommended). Cooperative Work Experience/Mathematics Cooperative work experience is an educational program that enables students to receive academic credit for on-the-job, experiential learning based on skills acquired in their programs. Together, the instructor, employer, and student establish learning objectives that specify the significant and appropriate learning which is expected to result from the work experience. This course offers a career-related experience for students working for an approved employer. As a capstone course, it should be completed within the last two terms of a certificate or degree program.
PUBLISHED PRODUCT TYPE 2,020Book 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. Show Less Read more Markov processes are used to model systems with limited memory. They are used in many areas including communications systems, transportation networks, image segmentation and analysis, biological systems... The only handbook of mathematical relations with a focus on particulate materials processing The National Science Foundation estimates that over 35% of materials-related funding is now directed toward... Praise for the First Edition: "For a beginner [this book] is a treasure trove; for an experienced person it can provide new ideas on how better to pursue the subject of applied statistics." —Journal... This introduction to Monte Carlo methods seeks to identify and study the unifying elements that underlie their effective application. Initial chapters provide a short treatment of the probability and... With an analytical approach that emphasizes concepts and comprehension, Data Matters provides a crucial introduction to statistics by preparing readers to think critically about the most common statistics... Provides readers with the methods, algorithms, and means to perform text mining tasks This book is devoted to the fundamentals of text mining using Perl, an open-source programming tool that is freely... Praise for the Second Edition: "This is quite a well-done book: very tightly organized, better-than-average exposition, and numerous examples, illustrations, and applications." —Mathematical Reviews... Praise for the Second Edition: "Serious researchers in combinatorics or algorithm design will wish to read the book in its entirety...the book may also be enjoyed on a lighter level since the different... "It is said if you take care of the pennies, the pounds will take care of themselves. Richard Burton's excellent book takes this approach to calculations applied to the biomedical sciences…This is certainly... An insightful approach to the analysis of variance in the study of linear models Linear Models explores the theory of linear models and the dynamic relationships that these models have with Analysis... The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present...
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Elementary Statistics - 11th edition Summary: Succeed in statistics with ELEMENTARY STATISTICS! With its down-to-earth writing style and relevant examples, exercises, and applications, this book gives you the tools you need to make the grade in your statistics course. Learning to use MINITAB?, Excel?, and the TI-83/84 graphing calculator is made easy with output and instructions included throughout the text. Need extra help? A wealth of online supplements offers you guided tutorial support, step-by-step video solutions, and imme...show morediate feedbackth Edition. Used - Good. Used books do not include online codes or other supplements unless noted. Choose EXPEDITED shipping for faster delivery! n $125.89 +$3.99 s/h New Premier Books BN Romulus, MI 0538733500 Brand New. Exact book as advertised. Delivery in 4-14 business days (not calendar days). We are not able to expedite delivery. $127.47 +$3.99 s/h Good invisibledog Salt Lake City, UT 0538733500 Instructor's copy Unmarked text. $139171215.51 +$3.99 s/h New Textbookcenter.com Columbia, MO Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book $225.75
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Rent Textbook Buy Used Textbook Buy New Textbook Currently Available, Usually Ships in 24-48 Hours $19.50 eTextbook We're Sorry Not Available More New and Used from Private Sellers Starting at $2Video Lectures on DVD with Optional Captioning for Intermediate Algebra through Applications Worksheets for Classroom or Lab Practice for Intermediate Algebra Summary This Geometry workbook makes the fundamental concepts of geometry accessible and interesting for college students and incorporates a variety of basic algebra skills in order to show the connection between Geometry and Algebra. Topics include: A Brief History of Geometry 1. Basic Geometry Concepts 2. More about Angles 3. Triangles 4. More about Triangles: Similarity and Congruence 5. Quadrilaterals 6. Polygons 7. Area and Perimeter 8. Circles 9. Volume and Surface Area 10. Basic Trigonometry Author Biography Alan Bass earned his Master of Science degree in Mathematics from the University of North Carolina at Wilmington. Presently, he lives, writes, runs, swims, bikes, hikes, and plays music with his wife Holly in sunny San Diego, CA. (He also enjoys fine wine and all things nerdy). He is an Associate Professor of Mathematics at San Diego Mesa College (just got tenure) and has been teaching developmental math for more than ten years. He is a big advocate for math study skills so, through a grant project called Pathways through Algebra, he got into networking with other schools to make study skills programs happen. He has worked on developing learning communities, establishing curriculum and pedagogy for study skills classes, and methods for incorporating study skills directly into the classroom. He has created and accumulated a ton of material, and wants to share it with you! For more information, check out his website at . Table of Contents Preface p. V A Brief History of Geometry p. 1 Basic Geometry Concepts p. 3 An Overview of This Supplement p. 3 Points, Lines, and Planes p. 3 Between Points p. 6 Congruent Objects and Bisectors p. 8 Solving Word Problems in Geometry p. 10 Construction Example: Using a Straightedge and Compass to Bisect a Line Segment p. 11 The History of Geometry p. 17 More About Angles p. 19 Measuring and Classifying Angles p. 19 Notation for Angle Measure and Congruence p. 21 Common Relationships between Angles p. 22 Parallel Lines Cut by a Transversal p. 27 Construction Example: Using a Straightedge and Compass to Bisect an Angle p. 29 Proofs in Geometry p. 35 Triangles p. 37 The Practical Nature of Triangles p. 37 Classifying Triangles p. 38 The Angles of a Triangle p. 41 Right Triangles: The Pythagorean Theorem p. 44 Comparing the Sides and Angles of a Triangle p. 45 Construction Example: Using a Compass to Create an Equilateral Triangle
Algebra I Algebra is generalized arithmetic, so basic math skills are reviewed and built upon throughout the course. Saxon math is used in the study of Algebra I and II. The Saxon approach uses incremental development and continual review. The incremental development is the introduction of topics in easily understandable pieces (increments), permitting the assimilation of one facet of a concept before the next facet is introduced. Both facets are then practiced together until another is introduced. All material learned is then reviewed in every lesson for the entire year. Topics are never dropped but are instead increased in complexity and practiced every day, providing the time required for concepts to become totally familiar. Math concepts steadily become more familiar and the requisite skills become automated. Learning is then demonstrated not only through the understanding of a concept, but also through the ability to apply that concept to new situations. Since Algebra I is the critical course for developing skills used in advanced math and science and for developing thinking skills used in modern technological society, this course involves rigorous practice.
Quick Links Algebra 2 (Michigan Edition) The content of Algebra 2 is organized around families of functions, including linear, quadratic, exponential, logarithmic, radical, and rational functions. In addition to its algebra content, Algebra 2 includes lessons on probability and data analysis as well as numerous examples and exercises involving geometry and trigonometry.
handbook (which includes an IBM-compatable diskette) aims to provide practising engineers with the principles and mathematical tools they need to perform sound economic analysis. Analytical methods covered include time-value, comparison of alternatives, break-even analysis, and risk analysis.
John Oliver Secondary Math 8 2010/2011 Math 8 Course Outline Welcome to Math 8! Teacher: Mrs. Burdett e-mail: mrsburdett@gmail.com website: Materials: (Bring to class EVERY DAY)  Text: MathLinks 8 and Mathlinks 8 Workbook ($10)  Calculator. Cell phones and iPods are note suitable, you MUST have a separate calculator.  2 inch Three-ring binder with 12 dividers: Your binder should be stocked with lined, unlined, and graph paper.  Pencil case: Your pencil case should contain pencils, eraser, blue (or black) and red pens, highlighters, metric ruler, and calculator.  Optional: cue cards, pencil crayons, felt markers, blank paper, scissors and glue stick.  Agenda: For keeping track op homework and assignment or test dates. You will be required to keep your agenda up-to-date. Marks: Assignments and Homework will be checked on a regular basis. Any student with incomplete homework or assignments may be required to stay after school and may not be allowed to write a test. Quizzes = 40% Chapter Tests = 60% Year Mark: Term 1 = 26% Term 2 = 27% Term 3 = 27% Midterm Exam = 10% of Term 2 Final Exam = 20% of your final grade Policies and Expectations: 1. Due Dates: all work is due at the beginning of class. Late homework may not be accepted. Assignments may be handed in late, but may lose marks. Once an assignment has been marked and returned, I may no longer be able to accept late ones. You may be required to complete an alternative assignment. 2. Absences: a note signed by a parent, guardian or physician must be handed in on the day you return. If you leave school partway through the day due to illness and are going to miss my class, you must inform me in person before you go. If you miss a test or quiz while you are absent, this note must have a physician's signature, or you may receive a zero. If you are sick enough to miss a test, you are sick enough to go to the doctor. YOU WILL MAKE UP A MISSED TEST OR QUIZ ON THE DAY YOU RETURN TO CLASS OR AT THE TEACHER'S CONVENIENCE. 3. Lates: a note or "admin slip" must be provided. No note? Your work will be considered late, and you will have a detention for double the time that you were late either at lunch or after school. There may be alternate consequences at the teacher's discretion. 4. Food: for safety reasons, there is no food, drink, or gum allowed in the science classroom. Water in a reusable container with a lid is acceptable when we are not doing a lab activity. 5. Respect:  Respect yourself by being the best person you can be. You are an intelligent, capable person. Participating, doing your homework and paying attention all serve your best interests.  Respect others by arriving on time and ready to learn (this means you have all materials with you and are in your seat when the bell rings). Listen attentively when others are speaking. Only one person speaks at a time, especially if that person is me! Absolutely No put downs or name calling.  Respect your environment by keeping the classroom clean. Put all recycling in the proper bin (paper flat, not crumpled). Don't leave anything in your desk and return clean lab equipment in the proper place. 6. Cheating and Plagiarism: Copying from others or from the internet, plagiarism, and cheating are unacceptable and will result in zero. This applies to ANY WORK handed in for ANY PART of Math 8. Refer to your agenda for the school policy on cheating. Students caught copying or allowing other to copy from them may receive zero and may not be allowed to make up the lost marks. Parents, all of you teachers, and the administration will be notified. This may result in suspension or other consequences to be determined by the administration. GRADE 8 It is expected that students will: NUMBER A1 demonstrate an understanding of perfect squares and square roots, concretely, pictorially, and symbolically (limited to whole numbers) [C, CN, R, V] A2 determine the approximate square root of numbers that are not perfect squares (limited to whole numbers) [C, CN, ME, R, T] A3 demonstrate an understanding of percents greater than or equal to 0% [CN, PS, R, V] A4 demonstrate an understanding of ratio and rate [C, CN, V] A5 solve problems that involve rates, ratios, and proportional reasoning [C, CN, PS, R] A6 demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially, and symbolically [C, CN, ME, PS] A7 demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically [C, CN, PS, R, V] PATTERNS AND RELATIONS Patterns B1 graph and analyse two-variable linear relations [C, ME, PS, R, T,V] Variables and Equations B2 model and solve problems using linear equations of the form ax = b   xa b , a ≠ 0 ax +b=c   xa b c , a ≠ 0 a(x + b) = c concretely, pictorially, and symbolically, where a, b, and c are integers [C, CN, PS, V]SHAPE AND SPACE Measurement C1 develop and apply the Pythagorean theorem to solve problems [CN, PS, R, V, T] C2 draw and construct nets for 3-D objects [C, CN, PS, V] C3 determine the surface area of right rectangular prisms right triangular prisms right cylinders to solve problems [C, CN, PS, R, V] C4 develop and apply formulas for determining the volume of right prisms and right cylinders [C, CN, PS, R, V] 3-D Objects and 2-D Shapes C5 draw and interpret top, front, and side views of 3-D objects composed of right rectangular prisms [C, CN, R, T, V] Transformations C6 demonstrate an understanding of tessellation by explaining the properties of shapes that make tessellating possible creating tessellations identifying tessellations in the environment [C, CN, PS, T, V] STATISTICS AND PROBABILITY Data Analysis D1 critique ways in which data is presented [C, R, T, V] Chance and Uncertainty D2 solve problems involving the probability of independent events [C, CN, PS, T
Synopses & Reviews Publisher Comments: Researchers, teachers and graduate students in algebra and topology--familiar with the very basic notions of category theory--will welcome this categorical introduction to some of the key areas of modern mathematics, without being forced to study category theory. Rather, each of the eight largely independent chapters analyzes a particular subject, revealing the power and applicability of the categorical foundations in each case. Synopsis: Synopsis: The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. "Synopsis" by Cambridge University Press,"Synopsis" by Cambridge University Press, The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure
013043 Mean Math Blues "A supplemental book" for courses in Study Skills. This text is designed so that reluctant and anxious math students learn current and relevant cognitive therapy and math study skill techniques. A broad variety of strategies journaling, self-assessment, goal setting, math exercises, questionnaires, webbing, etc. are designed to actively assist the student in pushing past their individual barriers to master math. Along the way, basic math exercises are introduced so that students can practice newly learned
Linear Algebra With Applications - 4th edition Summary: Linear Algebra with Applications is a flexible blend of theory, important computational techniques, and interesting applications. Instructors can select the topics that give the course their desired perspective. The text provides a solid foundation in the mathematics of linear algebra, while introducing some of the important computational aspects of the field, such as algorithms. The presentation of interesting applications has been one of the most compelling feature...show mores of this book provides students a well balanced coverage of standard linear algebra topics that apply mathematics by examining real-life applications, making for a enlightening learning experience71 +$3.99 s/h Good Greener Books London, 07/21/2000 Hardcover 4th Revised edition
regarde... read more Customers who bought this book also bought: Our Editors also recommend: An Introduction to Linear Algebra by L. Mirsky Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter. Applied Matrix Algebra in the Statistical Sciences by Alexander Basilevsky This comprehensive text offers teachings relevant to both applied and theoretical branches of matrix algebra and provides a bridge between linear algebra and statistical models. Appropriate for advanced undergraduate and graduate students. 1983 edition. Introduction to Matrices and Linear Transformations: Third Edition by Daniel T. Finkbeiner, II This versatile undergraduate-level text contains enough material for a one-year course and serves as a support text and reference. It combines formal theory and related computational techniques. Solutions to selected exercises. 1978 edition. Linear Algebra and Group Theory by V.I. Smirnov, Richard A. Silverman This accessible text by a Soviet mathematician features material not otherwise available to English-language readers. Its three-part treatment covers determinants and systems of equations, matrix theory, and group theory. 1961 edition. Theory of Sets by E. Kamke Introductory treatment emphasizes fundamentals, covering rudiments; arbitrary sets and their cardinal numbers; ordered sets and their ordered types; and well-ordered sets and their ordinal numbers. "Exceptionally well written." — School Science and Mathematics. An Introduction to the Theory of Linear Spaces by Georgi E. Shilov, Richard A. Silverman Introductory treatment offers a clear exposition of algebra, geometry, and analysis as parts of an integrated whole rather than separate subjects. Numerous examples illustrate many different fields, and problems include hints or answers. 1961 edition. Product Description: regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations. Table of Contents: l. The Algebra of Matrices 2. Linear Equations 3. Vector Spaces 4. Determinants 5. Linear Transformations 6. Eigenvalues and Eigenvectors 7. Inner Product Spaces 8. Applications to Differential Equations For the second edition, the authors added several exercises in each chapter and a brand new section in Chapter 7. The exercises, which are both true-false and multiple-choice, will enable the student to test his grasp of the definitions and theorems in the chapter. The new section in Chapter 7 illustrates the geometric content of Sylvester's Theorem by means of conic sections and quadric surfaces. 6 line drawings. lndex. Two prefaces. Answer section. Reprint of the Holt, Rinehart and Winston. Inc., New York, 1973
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Intermediate Algebra - 9th edition Summary: The Lial series has helped thousands of students succeed in developmental mathematics through its friendly writing style, numerous realistic examples, extensive problem sets, and complete supplements package. In keeping with its proven track record, this revision includes a new open design, more exercises and applications, and additional features to help both students and instructors succeed. Learning Objectives Each section opens with an application and clearly stated, numbered, learning objectives. These learning objectives are reinforced throughout the section by re-listing the objective where appropriate so that students always know exactly what is being covered. Classroom Examples and Teaching Tips The Annotated Instructor's Edition provides answers to all text exercises and Group Activities in color in the margin or next to the corresponding exercises. In addition, Classroom Examples and Teaching Tips are also included to assist instructors in coming up with examples to use in class that are different from what students have in their textbooks, and to offer guidance on presenting the material at hand. Notes Important ideas are emphasized in Notes boxes that appear throughout the text. Calling them out stresses their importance to students. Connections Connections boxes provide connections to the real world or to other mathematical concepts, historical background, and thought-provoking questions for writing or class discussion. Problem Solving The Lial six-step problem-solving method is clearly explained in Chapter 2 and is then continually reinforced in examples and exercises throughout the text to aid students in solving application problems. Cautions Students are warned of common errors through the use of Cautions that are found throughout the text where appropriate. Technology Insights Exercises Technology Insights exercises are found in selected exercise sets throughout the text. These exercises illustrate the power of graphing calculators and provide an opportunity for students to interpret typical results seen on graphing calculator screens (graphing calculator NOT required to complete these exercises). Ample and Varied Exercise Sets Algebra students require a large number and variety of practice exercises to master the material they have just learned. This text contains thousands of exercises, including review exercises, numerous conceptual and writing exercises, journal exercises, and challenging exercises that go beyond the examples. Multiple-choice, matching, true/false, and completion exercises help to provide variety. Exercises suitable for calculator use are marked with a calculator icon. Writing Exercises Writing exercises abound in the Lial series through the Connections boxes and also in the exercise sets. Some writing exercises require only short written answers, and others require lengthier journal type exercises that ask students to fully explain terminology, procedures, and methods, document their understanding using examples, or make connections between topics. Relating Concepts Found at the end of selected exercise sets, these exercises tie together topics and highlight the relationships among various concepts and skills. For example, they may show how algebra and geometry are related, or how a graph of a linear equation in two variables is related to the solution of the corresponding linear equation in one variable. These sets of exercises make great collaborative activities for small groups of students. Group Activities Appearing at the end of each chapter, these activities allow students to work collaboratively to solve a problem related to the chapter material. Ample Opportunity for Review One of the most admired features of Lial textbooks is the extensive and well-thought-out end-of-chapter material. At the end of each chapter, students will find Key Terms and Symbols that are keyed back to the appropriate section for easy reference and study, and Test Your Word Power to help students understand and master mathematical vocabulary. Key terms from the chapter are presented with four possible definitions in multiple-choice format. Quick Review sections give students not only the main concepts from the chapter (referenced back to the appropriate section), but also an adjacent example of each concept. Review Exercises are keyed to the appropriate sections so that students can refer to examples of that type of problem if they need help. Mixed Review Exercises require students to solve problems without the help of section references. Chapter Tests help students practice for the real thing, and Cumulative Review Exercises gather various types of exercises from preceding chapters to help students remember and retain what they are learning throughout the course. New To This Edition Former Chapter 5 Exponents and Polynomials has now been broken into two chapters: Chapter 5 Exponents, Polynomials, and Polynomial Functions and Chapter 6 Factoring. Division of Polynomials (formerly section 6.4, following Complex Fractions) has been moved up to Section 5.5 to follow Multiplying Polynomials. The Conics chapter (formerly Chapter 10, now Chapter 11) now includes a new section Additional Graphs of Functions; Composition. Determinants and Cramer's Rule (formerly Section 4.5) and Synthetic Division (formerly Section 6.5) have been moved to Appendixes B and C respectively. Adjunct Support Center The Adjunct Support Center offers consultation on suggested syllabi, helpful tips on using the textbook support package, assistance with textbook content, and advice on classroom strategies from qualified mathematics instructors with over 50 years of combined teaching experience. Now Try It Exercises To actively engage students in the learning process, each example now concludes with a reference to one or more parallel exercises from the corresponding exercise set so that students are able to immediately apply and reinforce the concepts and skills presented in the examples. Summary Exercises Based on user feedback, we have more than doubled the number of in-chapter summary exercises. These special exercise sets provide students with the all-important mixed review problems they need to master topics. Summaries of solution methods or additional examples are often included. Topics that appear in these Summary Exercises include solving linear and absolute value equations and inequalities, operations equations with rational expressions, and exercises on solving quadratic equations. Glossary By popular demand, a comprehensive glossary of key terms from throughout the text is included at the back of the book. New Real-Life Applications Many new or updated examples and exercises can be found throughout the text that focus on real-life applications of mathematics. These applied problems provide a modern flavor that will appeal to and motivate students. New Figures and Photos Today's students are more visually oriented than ever. Thus, the authors have made a concerted effort to add mathematical figures, diagrams, tables, and graphs whenever possible. Many of the graphs use a style similar to that seen by students in today's print and electronic media. Photos have been incorporated to enhance applications in examples and exercises. Marge Lial Marge Lial was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, is now affiliated with American River College. Marge is an avid reader and traveler. Her travel experiences often find their way into her books as applications, exercise sets, and feature sets. She is particularly interested in archeology. Trips to various digs and ruin sites have produced some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. Hornsby, John : John HornsbyWhen John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education, or journalism. His ultimate decision was to become a teacher, but after twenty-five years of teaching at the high school and university levels and ten years of writing mathematics textbooks, both of his goals have been realized. His love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history, and incorporating graphing calculators into the curriculum. John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recorded works of Frankie Valli and the Four Seasons. McGinnis, Terry : Terry McGinnis - A native Midwesterner, Terry received her Bachelor's of Science in Elementary Education (Mathematics concentration) from Iowa State University. She has taught elementary and middle school mathematics, and developed and implemented the curriculum used with her students. Terry has been involved in college mathematics publishing for over 15 years, working with a variety of authors on textbooks in developmental and precalculus mathematics. After working behind the scenes on many of the Lial/Hornsby textbooks and supplements for the past ten years, Terry most recently joined Margaret Lial and John Hornsby as co-author of their developmental mathematics series that includes Introductory Algebra, Intermediate Algebra, and Introductory and Intermediate Algebra, all published by Addison-Wesley. When not working, Terry enjoys spinning at a local health club and View Table of Contents List of Applications. Preface. Feature Walkthrough. 1. Review of the Real Number System. Basic Concepts. Operations on Real Numbers. Exponents, Roots, and Order of Operations. Properties of Real Numbers. The Square Root Property and Completing the Square. The Quadratic Formula. Equations Quadratic in Form. Summary Exercises on Solving Quadratic Equations. Formulas and Further Applications. Graphs of Quadratic Functions. More about Parabolas; Applications. Quadratic and Rational Inequalities. Additional Graphs of Functions; Composition. The Circle and the Ellipse. The Hyperbola and Functions Defined by Radicals. Nonlinear Systems of Equations. Second-Degree Inequalities and Systems of InequalitiesGoodwill Discount Books North Las Vegas, NV Good shape, medium wear. shows little to no wear $2.29 +$3.99 s/h Good TXTBookSales1 Evansville, IN Every book shipped with tracking number. May contain informative highlighting, markings, or cool sticker on the front. Overall pretty good shape. May not include Supplements, CDs or Access Codes. -Goo...show mored-
Cambridge popular introductory circuits text, known for its "learn-by-doing format" has been further improved with the additions of new problem-solving techniques and other learning enhancements. The presentations of the fundamental principles are replete with examples, drill problems, extension exercises and design problems. Editorial Reviews Booknews Textbook introduces the techniques of linear circuit analysis, primarily for electrical engineering majors. Requires math background through differential and integral calculus and an understanding of the techniques involved in the solution of differential equations with constant coefficients; basic knowledge of matrices and computer programming would be helpful 27, 2002 Not the greatest. This book gives examples which are much easier than the chapter problems. It does not give any answers for any chapter problems. The text refers figures which are almost always on a different page. Many examples have mistakes which makes the user unsure of their work. I consider myself an above average student and I have to obtain help to learn the topics. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
np. I rather people get it than just get an answer. That way you will know it when it is used in later math classes. For example, calculus is about 10% new stuff and 90% doing everything before calculus!
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Math 1A03 Calculus for Physical Science I Course Information Sheet Term 1 Autumn 2011 ... Calculus: earlytranscendentals, 7th edition, James Stewart, ... You may discuss the solution to a homework assignment with other students. You first, and then copy You can also use Stewart's Single Variable Calculus: EarlyTranscendentals (6th Edition) or Stewart's Calculus: EarlyTranscendentals (6th Edition), ... understand the solution to a problem before typing in your answer on WebAssign. Late homework assignments will not be accepted for ANY reason You can also use Stewart's Single Variable Calculus: Early Tran-scendentals (6th Edition) or Stewart's Calculus: EarlyTranscendentals (6th Edition), but you will ... sure you understand the solution to a problem before typing in your answer on WebAssign. Calculus I Fall 2013 Course website: ... Textbook: The textbook for this course will be Calculus, EarlyTranscendentals, 2nd edition by J. Rogawski. ... written assignments to understand a problem and to develop a solution. However, the nal Stewart OR CalculusEarlyTranscendentals, 7th ed. by Stewart You need chapters 10, 12, 14, and 15. Incomplete Grades: A Graphing Calculator ... is a student solution manual available for check out in the library, so you have access to the worked-out solutions to the odd- rst, and then copy –rst, and then copy University Calculus, Elements with EarlyTranscendentals by Joel Hass, Maurice Weir, and George Thomas, ISBN-10: 0-321-53348-8, ... Problems for which a serious attempt at a solution is not given will count against the completeness. All but the most minor of errors will Content: "Essential Calculus, EarlyTranscendentals," James Stewart, 2007 ... Related rates problems are easy if we follow a step-ways approach to solution: 1. Draw a picture and write down what you know and what you want to find. James Stewart, Calculus, EarlyTranscendentals, 6th edition, Cengage Learning, 2009. ... result contradicts common sense or the solution was suspiciously easily, then be wary. This is particularly important for tests and quizzes, but make a habit of it. Single Variable Calculus: EarlyTranscendentals, Seventh Edition James Stewart ... SOLUTION (a) We see from Figure 6 that the point lies on the graph of , ... that the ideas of calculus can be applied to a table of values; an explicit formula is not EXAMPLE 1 Evaluate the following limits and justify each step. (a) (b) SOLUTION (a) (by Laws 2 and 1) (by 3) (by 9, 8, and 7) (b) We start by using Law 5, but its use is fully justified only at the final stage when Single Variable Calculus: EarlyTranscendentals, Volume 2 Seventh Edition James Stewart ... Part 1 of the Fundamental Theorem of Calculus. Find . SOLUTION Here we have to be careful to use the Chain Rule in conjunction with FTC1. Let . ThenTextbook: University Calculus: Elements with EarlyTranscendentals, 1st edition, by Joel Hass, Maurice D. Weir, and George B. Thomas, Jr. ... Week 17 April 28 (M) Discussion and solution of Quiz 5 April 30 (W) Last day of class, Review for final exam Textbook: We will be using Calculus: EarlyTranscendentals by Briggs and Cochran. You may nd the book in the bookstore. Alternatively, you may ... a task, problems requiring you to set up but not evaluate the solution to a problem, and computational problems asking you to work though a problemfrom the prescribed text book – Calculus, EarlyTranscendentals, Briggs, Cochrane First order linear differential equations ... In addition, the student will give a presentation of the solution of a problem chosen from a list of problems that will be posted on Blackboard around Spring break. Text: Single Variable Calculus: EarlyTranscendentals (6th edition), by James Stewart ... • You should write up your solution separately from your collaborators and without reference to the white board, blackboard, or sheet of paper containing your joint work so as to com- Calculus - EarlyTranscendentals, Sixth Edition, by Stewart. Course Website: ... Even if your solution is correct, expect to lose points if it is difficult to read and understand. This includes solutions that are confused, incomprehensible, ... Text: Calculus, EarlyTranscendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. ... The computer generates problems and guides you to the solution. We will not keep track of your performance on WebAssign. If you purchase your text at the Lehigh Bookstore, part of the price Math 2210 Multivariable Calculus Name Homework 2: The Dot Product Due: January 15 at the beginning of class Directions: Write your solutions to the following problems on notebook paper with clean edges solution of any problem.Your problem may be modest;but if it challenges your curiosity and brings into play your inventive faculties,and if you solve it by your ... Calculus, EarlyTranscendentals, Sixth Edition,is supported by a complete set of ancil- Math 2210 Multivariable Calculus Name Homework 4: Lines and Planes Due: January 24 at the beginning of class Directions: Write your solutions to the following problems on notebook paper with clean edges
You are here Interactive Internet Based Multivariable Calculus INTERACTIVE INTERNET-BASED MULTIVARIABLE CALCULUS Thomas Banchoff June 26-29, 2005 Brown University Providence, RI Visualizing graphs of functions of two and more variables is a key skill needed to deal with the main ideas of multivariable calculus. Internet-based tools make it possible to engage students in the interplay between analysis, algebra, and geometry in ways that lead to deeper appreciation of the underlying principles and of real-world applications. New software developed by the workshop leader and his students provides accessible and powerful tools for interactively investigating curves and surfaces, and enables students to share their own demonstrations with the instructor and with other students. Communication software provides easy ways for students to respond online to homework assignments and even examinations, and for instructors to submit online comments that can be shared, after a time delay, with other members of the class. One student, describing her experience in the class, wrote: 'In the beginning I really did not like the system of reading other students' homework. It made me feel really nervous and exposed and it made me want to leave problems blank rather than put an incorrect answer. As the semester progressed I realized what a useful tool it could be and I started reading other people's homework responses more and more and felt more comfortable with mine being read.� This workshop is intended for teachers of multivariable calculus who wish to explore the possibilities of new software for helping students develop a full range of geometric intuitions as well as formal skills in algebra, in abstract concepts in calculus (including continuity, differentiability, and curvature) and in effective writing. Prior computer experience is not required. The workshop will use Java applet software and laboratory materials that will be available for the participants to use in their courses in the fall. In addition to the three-day intensive workshop, there will be online follow-up activities at least once a month during the fall, concentrating on different ways that the participants experiment with the software in their courses. Members of the workshop will be invited to participate in a session at the joint MAA-AMS national meeting in January 2006 to share their experiences. For more information, please visit the workshop webpage at
Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to provide an introduction to the elementary complex transcendental functions -- the exponential, sine, and cosine functions.... Graph is "an open source application used to draw mathematical graphs in a coordinate system." Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to visualize a... Once again, the Mathematical Association of America has struck instructional gold with this latest gem from their online collection of resources for mathematics educators. Created by Barbara Margolius, this derivative... Created by Illumination: Resources for Teaching Mathematics, this unit plan includes four lessons covering correlation and regression. Students will use interactive computer-based tools to investigate the relationship... Created by Illuminations: Resources for Teaching Mathematics, this lesson plan uses the Birthday Paradox to introduce basic concepts of probability. Students run a Monte Carlo simulation using the TI-83 graphing...
Algebra City Focuses on the 28 Most Common Misconceptions about Algebra as Part of Assessment-Driven Intervention SAN ANTONIO, Feb. 2, 2012 /PRNewswire/ -- With many states requiring Algebra I to graduate from high school, algebra has become one of the gateway courses to school and career success. Yet upwards of 60 to 70 percent of students struggle with algebra or fail to pass state-mandated proficiency exams. PCI Education, the premier provider of resources for students with specialized instructional needs, introduces Algebra City™,a blended intervention program focusing on the 28 most common algebraic misconceptions. Research shows that many students misunderstand the concepts, procedures and representations needed to master and pass Algebra I. Algebra City aims to keep students on track by using pinpoint assessment to identify where a student is struggling conceptually, and providing thorough and multiple approaches to correcting the misconception. Algebra City may be used for intervention with any core Algebra I curriculum. According to Algebra City author Dr. Donna Craighead, the program's four Student Editions differ from traditional algebra textbooks. Whereas textbooks use a linear model, as an intervention program Algebra City uses assessment data to target instruction only where needed. The graphic novel-style Student Editions use avatar-like characters to encourage students to re-engage with algebra in new and exciting ways, including an online adventure island where students can solve practice problems. Aligned to the Common Core State Standards, Algebra City is a four-part series, with each book covering seven misconceptions. The series is divided into Algebra Essentials, Equations & Inequalities, Graphing, and Polynomials & Factoring. The ExamView Assessment Suite for Algebra City includes readymade pre- and post-tests at the program, book and unit levels, an item bank and test generator, and robust reporting. "Too often, students struggle to learn critical algebra skills they need both inside and outside the classroom," said Lee Wilson, president and CEO of PCI Education. "Algebra City is targeted intervention that encourages students to reconnect to algebra in one or more areas of misunderstanding, while allowing teachers to leverage the investment in their core algebra curriculum." Algebra City is one of five new offerings from PCI Education that provide intensive intervention and remediation in reading, writing, and math for students in grades 6-12. About PCI Education PCI Education offers more than 7,500 educational materials for a wide range of students with specialized instructional needs. The company's products are used to help students performing below grade level, students with learning differences, and students with significant or developmental disabilities such as autism. In addition, PCI programs are used in English language learner and adult literacy classes. Based in San Antonio, PCI Education has been helping educators lead students to success in school, at home and in the community since 1991. For more information, visit
Site Navigation: Onward Courses, Labs and Degree Credit - Mathematics Overview Onward Mathematics courses are designed to give you the basic skills needed for success in future math courses at UMaine and for many personal and professional situations as well. Credit for these courses will not count towards your bachelor's degree (except for ONM013, see note), but they do count for financial aid, and all UMaine add/drop and withdrawal policies apply. This course covers the basic topics in algebra needed to enter a mathematics course at the precalculus level. The covered topics include a brief review of the real number system (including absolute value, exponents, roots, and radicals), linear equations and inequalities, quadratic equations, graphs, functions (primarily linear and other polynomial), factoring, rational and radical expressions. Note: This course counts for 3 degree credits, but it does not satisfy UM's General Education Mathematics Requirement. Prerequisite: ONM 12 or permission. 3 degree credits.
Barron's Regents Exams and Answers: Math B 1 rating: 5.0 A book by Lawrence S. Leff This brand new Regents review "Redbook" conforms to the new Regents Math B curriculum taught in secondary schools throughout New York State. The new Math B exam covers topics that until recently were included in portions of the old Sequential Math Course … see full wiki Let's Review Math B by Barron's This text has been updated through 2006 including the 2005 Regents examinations. The newer volume has fairly comprehensive proofs of the parallelogram, trapezoid and other elusive mathematical forms. For instance, the parallelogram is proven; thusly, - congruency of vertical angles - opposite sides are parallel - when 2 lines are parallel, the alternate interior angles are = - midpoint referencing equal segments which are parallel length are congruent There is an extensive section on arithmetic operations, complex numbers, translation points, trigonometry, quadratic models and solution sets for recent regents examinations. The work is easy to understand, it is current through to 2006 and the price is reasonable.
0199230722 9780199230723 General Relativity and the Einstein Equations:General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field. Back to top Rent General Relativity and the Einstein Equations 1st edition today, or search our site for Yvonne textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Oxford University Press, USA.
Detailed Course Information When the term course schedule is available the "Schedule Types" are hyperlinks. Clicking on the link will then display individual section information. MATH 050 - PreAlgebra This course reviews mathematical operations involving fractions and decimals. Topics include percents, ratios, proportions, U.S. and metric measurements, integers, statistical graphs, Pythagorean Theorem, perimeter, area, and volume. It also introduces algebraic concepts using expressions and equations. Problem solving, estimation, and reasoning skills are taught. The calculator and real-life applications are integrated throughout the course. (F,Sp,Su) Prerequisite: Math Level 3 and Reading Level 3 and Writing Level 2 Course Note: MATH 050 is usually offered in 4 different delivery methods: Online, Online/Hybrid, Lecture, Learning Lab. For an explanation of the Learning Lab delivery method, refer to select What is a Learning Lab?, or call (517) 483-1073 and press 4. The course fee provides all enrolled students online access to a full electronic textbook (e-text), online course homework system, and additional online resources. These course materials will be available to students the first day of class. Students are encouraged to speak with their instructor BEFORE purchasing any optional course materials. The TI-30XIIS is required for course work. A graphing calculator from the TI-83/84/Nspire family would also be acceptable.
More About This Textbook Overview Includes 21 competencies/skills found on the TExES Mathematics 8-12 test and 125 sample-test questions. This guide, aligned specifically to standards prescribed by the Texas Department of Education, covers the sub-areas of Number Concepts; Patterns and Algebra; Geometry and Measurement; Probability and Statistics; Mathematical Processes and Perspectives; and Mathematical Learning, Instruction and AssessmentRecommend with reservations Best out there, but the language is very technical and hard to follow; very dry. If you were a soft science major, like me (Psychobiology, UCLA), you're in for a long trip. Good luck. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted June 7, 2007 Be ready to proofread I would have given this a 3 except this is the best resource I have found. You need to understand what you are looking at because there are several typo's and I have found 1 answer to be incorrect. If you catch the mistakes, you know what you are doing and this is a great review. If not, I would consider a different resource, possibly some old used textbooks. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
EDUC 321 Mathematics Curriculum, Methods, and Materials Prerequisite: Admission to Phase II. (Both semesters/3 credits) An examination of modern materials and methods through a constructivist-based instructional approach. Emphasis is placed on the study of current school practices and the implementation of national and state standards. Methods emphasize appropriate activities at the concrete (manipulative), pictorial and abstract levels. Curricular topics include mathematics as problem solving, communication, reasoning and making connections with the learner's world
Mathematical Modelling in One Dimension demonstrates the universality of mathematical techniques through a wide variety of applications. Learn how the same mathematical idea governs loan repayments, drug accumulation in tissues or growth of a population, or how the same argument can be used to find the trajectory of a dog pursuing a hare, the trajectory of a self-guided missile or the shape of a satellite dish. The author places equal importance on difference and differential equations, showing how they complement and intertwine in describing natural phenomena.
Re: Systems of Linear Equations Wonderful! If they would only introduce the subject in this manner. They did not in my time. Judging from the fact that so many students do not understand anything about simultaneous equations... The three diagrams "Inconsistent", "Independent", "Dependent" are the fundamental geometric interpretation by which half of Numerical Analysis can be understood. Because these concepts are usually introduced through determinants the concept of ill conditioned cannot be understood correctly. Didn't even bother to look for errors. Loved the page. In mathematics, you don't understand things. You just get used to them. Some cause happiness wherever they go; others, whenever they go. If you can not overcome with talent...overcome with effort.
The Logic of Long Division Overview This book teaches the mechanics and methodology of long division, a procedure for dividing numbers without the need for an electronic calculator. Starting with basic concepts, the book explains the method step by step, and then reinforces these concepts using extensive examples and problems with complete solutions. A Tarrington Math Series Book. Most appropriate for grades 5 to 858129376 There are no customer reviews available at this time. Would you like to write a review?
From an extraordinary converted convent in the romantic English countryside, to the art-filled Manila sanctuary of a renowned gallery owner, the second edition of Philippine Tatler Homes is a delightful compilation of some of the world's most beautiful houses. The Chinese migration to the Latin America/Caribbean region is an understudied dimension of the Asian American experience. There are three distinct periods in the history of this migration: the early colonial period (pre-19th century), when the profitable three-century trade connection between Manila and Acapulco led to the first Asian migrations to Mexico and Peru; the classic migration period (19th to early twentieth centuries), marked by the coolie trade known to Chinese diaspora studies; and the renewed immigration of the late 20th century to the present. Written by specialists on the Chinese in Latin America and the Caribbean, this book tells the story of Asian migration to the Americas and contributes to a more comprehensive understanding of the Chinese in this important part of the world. Designed for the non-traditional Liberal Arts course, Mathematical Thinking and Quantitative Reasoning focuses on practical topics that students need to learn in order to be better quantitative thinkers and decision-makers. The author team's approach emphasizes collaborative learning and critical thinking while presenting problem solving in purposeful and meaningful contexts.

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