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USEFUL LINKS:
STUDENT LEARNING OUTCOMES
Upon successful completion of Math 250, the student
should be able to:
1. State and apply basic definitions, properties,
and theorems of first semester Calculus.
2. Model and solve problems using derivatives of algebraic and transcendental
functions.
3. Analyze and sketch graphs using the principles of calculus.
4. Evaluate limits, derivatives, definite and indefinite integrals graphically,
algebraically, and/or using the formal definitions.
STUDENT LEARNING OBJECTIVES:
Student will review the concepts of absolute
value inequalities, functions, combinations of functions and composite functions.
Student will evaluate limits using the limit
theorems, the definition, and demonstrate the concept of continuity.
Student will find the derivative (first and
higher order) using rules, theorems and definition. Interpret as slope, rate
of change, velocity, and acceleration.
Student will learn and demonstrate techniques
of approximation, and the differential. |
For a subject that is a challenge at all levels of education, this chart covers principles for basic algebra, intermediate algebra and college algebra courses. Topics covered include: set theory operations of real numbers algebraic terms steps for solving a first-degree equation with one variable steps for solving a first-degree inequality with one variable order of operations factoring special factoring hints rational expressions complex fractions synthetic division roots & radicals rational expressions in equations radical operations radical expressions in equations quadratic equations complex numbers |
Mathematical Techniques: An Introduction For The Engineering, Physical, And Mathematical Sciences
Mathematical Techniques: An Introduction For The Engineering, Physical, And Mathematical Sciences62
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Book Summary of Mathematical Techniques: An Introduction For The Engineering, Physical, And Mathematical Sciences
Many students beginning their engineering, science and mathematics courses need a book on mathematical methods. This textbook offers an accessible and comprehensive grounding in many of the mathematical techniques required in the early stages of an engineering or science degree, and also for the routine methods needed by first and second year mathematics students. "Mathematical Techniques" starts by revising work from pre-university level before developing the more advanced material which students will encounter during their undergraduate studies. The contents of the book has been fully revised for this, the third edition. The first chapter on standard techniques, has been rewritten and expanded to serve the increasingly diverse needs of students. The Fourier transform now has its own chapter; a simplified approach is adopted, and diffraction theory, together with supporting material on wave motion, is included. Many changes enhancing clarity have been made in other chapters. The chapter on projects using Mathematica has been extended to cover these changes: the associated programs are freely available on Keele University, Mathematics Department web site. Chapters and sections are designed to be largely self-contained, allowing students to concentrate on the specific methods they need to master and use. The book contains nearly 500 worked examples, more than 2000 problems (with selected answers), and over 120 computing projects. The text is accessible, widely illustrated, and stands as an ideal introduction on mathematical methods at university level.
Books Similar to : Mathematical Techniques: An Introduction For The |
This course provides basic skills in intermediate algebra. Topics: system of linear equations in two or more variables, trigonometry of the right triangle, radicals, the system of complex numbers, graphs of conic sections, and graphs of trigonometric functions.
This course is designed for Allied Health majors and will aid them in applying mathematical concepts to on-the–job situations. The course will include: an integrated review of the mathematical skills required in Allied Health Professions, in particular those topics pertaining to Pharmacology and Radiology.
The student will verify some fundamental properties of natural numbers, express numbers in different bases, find the greatest common divisors of two numbers by Euclid's algorithm, factor an integer by various methods such as Fermat's and Euler's methods, and become acquainted with several solved and unsolved problems in number theory. The student will find the number of divisors of a natural number, the sum of the divisors, the product of the divisors, and the means of the divisors; become acquainted with perfect, multiple perfect, amicable and sociable numbers; analyze various theorems related to perfect numbers; study Euler's function; solve simple diophantine equations; and study congruences.
MAT 120
INTRODUCTION TO PROBABILITY & STATISTICS
3 credits, 4.5 hours
Prerequisite: Passing score on the Compass Pre/Corequisite: ESL 35
The student will explore, describe, and compare data by measures of central tendency and dispersion from selected sample data sets. Using the sample statistics, the student will be able to make a statement about the population parameters by confidence-level and hypothesis testing methods. The student will also solve problems involving probabilities ands their distributions. Other topics such as correlation, regression, chi-square and analysis of variance will also be covered.
The student will study the following as they relate to computers: the algorithm, its expression as a flowchart, a computer model and a computer language (BASIC), computation of a data organization, arithmetic expressions, compound conditions, branching, arrays, and looping. The student will also study the following as they relate to computers: approximations, functions and procedures, numerical applications, roots of equations, maxima and minima, areas, simultaneous equations, averages and deviation from the average.
MAT 160
PRE-CALCULUS
4 credits, 4.5 hours / .5 hr lab
Prerequisite: MAT 30 AND passing score on the Compass or initial placement through the Compass
This course provides essential concepts for the study of calculus. Topics: concepts in analytic geometry; algebraic functions; transcendental functions, such as exponential, logarithmic, and trigonometric functions; graph analysis; and applications.
This course provides skills in differential and integral calculus. Topics: definite integral and its properties, numerical integration, applications of the definite integral to areas, solids of revolution and length, inverse functions, logarithm and exponential functions, conic sections, and translation and rotation of axes.
The student will study VECTOR CALCULUS, matrix algebra, system of homogeneous and non-homogeneous linear equations, concepts of vector space, subspace, basis and dimension of a vector space, linear transformation, and Eigenvalues and Eigenvectors for a linear transformation.
The student will formulate and solve differential equations of the first and second order. She/he will apply these methods to related practical problems. The student will formulate and solve linear differential equations with constant coefficients and apply these techniques to practical problems that give rise to such equations. |
Not only helpful with math, but that's what I use it for. You can type in pretty much any kind of problem and it will solve it, or at least give you helpful information that will lead to the solution. I'm taking an online college algebra class at the moment and I could probably pass the class using nothing but this.
EDIT: This is also really helpful, often even more helpful than WolframAlpha:
Wolfram alpha is REALLY overkill for all high school math. It really should only be used for calculus, and even with calculus it has trouble with large integrations and derivatives.
Personally I don't suggest you use Wolfram Alpha unless you already know how to do the math and just need it to reduce a ridiculously complex problem. Otherwise you're gimping yourself in the long run by relying on a piece of technology to do your work instead of knowing how to do it yourself.
Besides, for simple algebra it only takes a few seconds to solve in your head. It takes 7 times as long to type that into wolfram alphaA generic graphing calculator can do everything you need to know in math class up until calculus. Until that point, using Wolfram Alpha is overkill. The only upside to it I can think of at this moment is the additional notation that you learn from using Wolfram Alpha.
etc. Their full, pay-to-download program (Mathematica) is great as well, but the notation is a bitch to learn.
The notation is just LaTeX. If you're going to do anything in the sciences or maths, you'll need to learn LaTeX anyway.
Mathematica scripting is also pretty strait forward and very user friendlyEssentially, I'd liken Mathematica to photoshop - it's friendly enough for simple and familiar computations, but there's a certain amount of complexity inherent in its versatility. I probably know about 5% of the commands, as a generous estimateNot really sure if on topic or not, but:
For anyone wanting to use those Wolfram addons, but with difficulties with the notation - you could always mess around with the online equation editor: It shows what the formula gives, and the source as well. |
Background for Digital Filing CabinetsThe world is making progress toward achieving free, universal elementary school education. This, of course, is merely a step toward providing free PreK-12 or PreK-16 or lifelong education for all people of all ages throughout the world. Information and Communication Technology (ICT) is making a steadily growing contribution toward eventual achievement of these visionary goals.
Open Source Textbooks. This Web Page explores the idea of providing free, open source textbooks and instructional materials to students took keep, edit, add marginal notes and comments, and so on.
Introduction to Math Education DFC
Each academic discipline has its own discipline-specific educational goals and ways of achieving these goals. In our current school curriculum, it is useful to think of how to improve education in specific disciplines. This Web Page focuses specifically on the idea of a Math Education Digital Filing Cabinet.
It is important to remember, however, that most problems people encounter are interdisciplinary. Math is an important aid to representing and attempting to solve problems in every academic discipline. Thus, math needs to be taught in a manner that facilitates transfer of learning to other disciplines, and other disciplines need to be taught in a manner that helps students learn to make effective use of math in the disciplines.
The Math Education Digital Filing Cabinet project is based on three assumptions:
That all people of the world are entitled to a free, good quality education. Good quality is to be determined by contemporary standards; however, it should prepare students to become and remain responsible citizens and lifelong learners who can adjust to life in a changing world.
This education should be designed to empower learners by helping them gain levels of expertise in diverse areas that meet their own specific needs and interests, the needs and interests of their community, and the needs and interests of the world.
Knowledge and skills in math and in using math to help represent and solve problems are an important outcome of a good education.
The Math Digital Filing Cabinet project is very large and is just in its infancy. This IAE-pedia page is being used to explore various aspects of the project. Eventually there will be separate Digital Filing Cabinet drawers for various groups of math teachers. For example, the needs of an elementary school teacher are quite different than the needs of a College of Education Math Methods teacher or a university Department of Mathematics faculty member who provides math content instruction to preservice elementary and secondary school teachers.
Math Education for Teachers of Math
Math is a broad, deep discipline with a long history. A person can spend a lifetime studying and doing research on math content and still know only a small fraction of the totality of collected math knowledge. Similar statements hold for a person exploring the history of math, the teaching of math, and the applications of math in various non-math disciplines.
A person who is teaching math or teaching teachers to teach math needs to be knowledgeable in three overlapping areas of mathematics:
Math content knowledge.
Math pedagogical knowledge.
Math pedagogical content knowledge (PCK).
The diagram given below is applicable in every academic discipline.
Math Content Knowledge
There is a huge and steadily growing accumulation of math content knowledge. On a worldwide basis, many thousands of math researchers are contributing to this accumulation.
The challenge of this huge and steadily growing accumulation of math content knowledge can be examined from how it affects elementary teachers, secondary school math teachers, and higher education math teachers. The situation is roughly as follows:
A typical elementary school teacher has studied math up through the 11th or 12th grade, has taken a Math For Elementary Teachers course or sequence of courses in college,and has taken a Math Methods course. This persons "peak" math content is the material covered in Math for Elementary Teachers, which may have College Algebra as a prerequisite.
A typical secondary school math teacher has a math content preparation that lies in the range of two years of college math to a bachelor's degree in math. In a number of states, there is a strong emphasis on high school math teachers having a bachelor's degree in math.
A typical teacher of math in a higher education institution has math content preparation that lies someplace in the range of a bachelors degree in math to a Doctorate in math.
For most teachers of math, there is a considerable difference between their highest level of math content course work and their current level of math content knowledge and skill. On the one hand, we know that people tend to forget the details of coursework that they are not using on a regular basis.Thus, for example, a typical fourth grade teacher will gradually forget most of the details of math content learned in high school and above.
On the other hand, the research-oriented math faculty in a college of university university will be routinely actively engaged in maintaining and expanding their math content knowledge. Thus, especially in their areas of research, their content knowledge will be well above the level achieved while in school.
Math Pedagogical Knowledge
A relatively strong rule of thumb is that teachers teach the way they were taught. The pedagogical knowledge gained by years and years of observing teachers (being taught by teachers) create a powerful mind set on how teaching is done.
Think about your experiences as a math student in elementary school, in secondary school, and in college. During these years of your schooling, you learned how elementary teachers typically teach math, how secondary school math teachers typically teach math, and how college math teachers typically teach math.
In secondary school, for example, the math class might begin with students handing in an assignment that they started working on during the previous math class. This is followed by a discussion of assignment problems, presentation of some new material, a new assignment, and seat work for the remainder of the period. The class may well include students doing some work at the chalk board, and the teacher will likely use an overhead projector, computer projector, chalkboard, or white board in the presentation. The amount of interaction between the teacher and students may vary considerably depending on the students and the teacher. Some teachers may have students interact in small groups to explore a problem of mathematical task.
In college math courses, demonstration and lecture tend to dominate. The teacher demonstrates and explains how to solve various problems that were in the homework assignment. The teacher lectures and demonstrates on the new material to be presented. Students take notes, and they ask questions about parts of the demonstration and lecture that they do not understand. From time to time the teacher asks a question and accepts ananswer from some volunteer in the class.
A student in a preservice teacher eduction program has repeatedly seen examples of math teaching and has gained quite a bit of math pedagogical knowledge. Research indicates that teachers tend to teach the way they were taught. It is hard for a teacher at any level to break the math pedagogical knowledge patterns they grew up with.
This creates an interesting and large challenge to the math education community as research suggests new and possible better ways to facilitate student learning.
Here is an example. Consider the idea of student-centered teaching, small group discussions, and team projects in teaching and learning math. How does a preservice teacher who has seldom or never participated in such teaching/learning environments learn to make effective use of these teaching techniques?
For another example, how does one make effective use of a computer hooked to a projection system and to the Internet while teaching a math unit of study? At the current time, relatively few students are seeing good examples of this in their elementary school secondary school, and college math courses. The idea of virtual manipulatives is related to this. Many elementary school math teachers are comfortable with students using physical manipulatives. What are advantages disadvantages of using computer-based manipulatives?
Math Pedagogical Content Knowledge
The idea of pedagogical content knowledge (PCK) has received a lot of attention and has been the focus of quite a bit of research and teacher education since it was first proposed by Lee Shulman in the mid 1980s. Quoting from the [ Technology Pedagogical Content Knowledge Website:
This knowledge includes knowing what teaching approaches fit the content, and likewise, knowing how elements of the content can be arranged for better teaching. This knowledge is different from the knowledge of a disciplinary expert and also from the general pedagogical knowledge shared by teachers across disciplines. PCK is concerned with the representation and formulation of concepts, pedagogical techniques, knowledge of what makes concepts difficult or easy to learn, knowledge of students' prior knowledge and theories of epistemology. It also involves knowledge of teaching strategies that incorporate appropriate conceptual representations, to address learner difficulties and misconceptions and foster meaningful understanding. It also includes knowledge of what the students bring to the learning situation, knowledge that might be either facilitative or dysfunctional for the particular learning task at hand. This knowledge of students includes their strategies, prior conceptions (both "naïve" and instructionally produced); misconceptions students are likely to have about a particular domain and potential misapplications of prior knowledge.
Liping Ma is well known for her 1999 book Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the U.S. Her book provides good examples of math PCK needed by elementary school teachers. She argues that even though elementary school math teachers in China have had quite a bit less formal math instruction than similar teachers in the United States, the Chine teachers have better PCK because their elementary and secondary school teachers had better PCK.
Information and Communication Technology
Information and Communication Technology (ICT) is affecting the content, pedagogy, and PCK of every academic discipline. Here is a very brief summary of the current and future situation:
Computers can solve or significantly help to solve some of the problems in each academic discipline. The ICT content in any particular discipline varies with the discipline. However, ICT is now important enough in each discipline so that the content being taught to students needs to reflect capabilities and limitations of ICT in the discipline.
ICT provides a variety of pedagogical aids. For example, computer-assisted learning and distance learning are of growing importance in each academic discipline.
Teachers in each discipline are faced by the challenge of how to help students learn to use ICT as an aid to knowing and using the discipline being taught. ICE brings some general aids to teaching and learning, such as computer-assisted learning, distance learning, and use of multimedia in classroom instructional. Each specific discipline has its own ways of dong this and its own discipline-specific materials that are relevant to such tasks.
Math games and puzzles, along with some information about suitable uses to improve the quality of math education that students using the games and puzzles are apt to gain.
Archival copies of grade books from past years.
A library of books, magazines, journals, and articles to support personal needs and needs of one's students.
Overhead projector foils. There are lots of different possibilities. For example, many teachers find it useful to have various sizes of black line masters of graph paper. Of course, blank foils and appropriate marking pens are a needed part of this collection.
Physical manipulatives. Examples include blocks, spinners, geoboards, dice, and so on. Teachers making use of class sets of such manipulatives need a lot of drawer and/or shelf space.
Video tapes, audio tapes, CDs, and DVDs. In each case, a teacher may have both prerecorded and blank recordable media.
Storage boxes and containers.
Papers and tests received from and/or ready to be handed back to students.
Etc. etc. etc.
Some items might be stored in either a physical filing cabinet or a DFC, or both. Some possible examples include grade books, quizzes and tests, lesson plans, handouts for use by students.
A physical filing cabinet might consist of some materials stored at home, some stored in one's classroom, and some stored elsewhere, such as a storage room in one's school. ideally, all of the contents would be easily accessible when you want to access them. This is an obvious problem with the storage and retrieval of physical materials. As an example, usually one wants to avoid the cost of having duplicate copies of reference books that one might want to refer to at home and at school.
With physical materials, there is an ongoing problem of guarding against possible disasters. What happens if there is a fire or flood in the space where your materials are stored? What happens if materials are stolen or maliciously destroyed? If a teacher moves to a new teaching job, who gets to keep the physical materials?
Here is an important question. As the collection grows, how does one organize it so that needed materials are quickly retrieved? Here are two interesting aspects of an answer.
First, one only tends to collect and store materials that are specifically relevant to the job. One has a good working knowledge of how to use the materials on the job. Thus, one has a type of personal "ownership" of the materials.
Second, the materials are often stored in a manner so that one's kinesthetic sense, visual memory, and a quick glance tend to help in quick retrieval. It is an interesting exercise to compare and contrast this with retrieval of information stored in a computer.
Math DFC: Where to Put It?
Anything that can be stored in a computer can be part of your DFC. A major question is, where do you want to store your Math DFC contents. Two general choices are:
On your own personal computer or computers.
On a server. This might be a local server, such as in your school building or district. It might be a server located thousands of miles away. (Indeed, you may have no idea where your materials are being stored.)
Nowadays, you are apt to have part of your DFC on your own personal computer and part on servers. In either case, you need to be concerned with having the material regularly backed up in a "off site" location and having your materials protected from physical and electronic threats. You also need to decide what parts of your DFC you want to make available to other people.
Suppose, for example, that all of the contents of your DFC are stored on a laptop computer that you regularly carry between home and school. You might have a home desktop microcomputer where you keep a backup copy of your DFC. If your storage and backup are just on these two computers, have the risk of both computers being damaged at the same time by fire, storm, or flood. Thus, you still need to have some form of off site backup storage.
When you want to share a document with a particular person you can send it as an email attachment, copy it onto a CD or DVD and hand the person the physical medium, copy the material onto a thumb drive and watch as the person copies it into his or her computer, and so on. In all of these cases you have considerable control over who receives the material. Of course, you don't know who might get copies from this person.
The Information Age Education (IAE) Websites provide an example of storage on a server. The company running the server provides automatic backup. In addition, Information Age Education staff periodically make an off site backup.
The Web Page you are currently reading is stored on a server, along with the MediaWiki software that is used in creating, editing, storing, and retrieving the material. All of the content can be accessed from any place in the world by anyone who can access the Internet. Most of the pages in the Website are open to online editing by readers. That is, people from around the world can log onto the site, add and/or edit pages, add comments, and so on.
The content stored on the Website (along with Joomla! software) is open to reading and download by people throughout the world. However, the content posted on this Website cannot be changed by the readers.
A third alternative available when storing on a server is to have the contents password protected. Then, only people who know the password can log on and access the content.
In summary, storage on a server run by a reputable and responsible organization provides for off site backup and for ease of access by people throughout the world. Access to the content may be restricted through use of some sort of password protection system.
Math DFC: Making It Yours
It does little good to collect lots of stuff into a DFC and have no idea what is there or how and why you might want to use it some day. You need to have personal ownership and understanding of the content of your DFC.
Today's search engines make it relatively easy to retrieve thousands (indeed, millions) of articles on various topics that are relevant to teaching and learning math. Each time you find an article of personal use to you, think about adding it to your DFC. One way to do this is via an annotated bibliography. Put an entry into the "references" section of DFC that contains a proper citation to the article, including a link to its website if the article is on the web. Then write a brief paragraph (in essence, a note to yourself) explaining what this article means to you and why or how you might want to use it in the future.
Teachers are used to the idea of developing lesson plans, using the lesson plans, and writing comments to themselves about what they will do the same and what they will do different the next time they use the lesson plans. This is an excellent example of steadily increasing the value of material in one's filing cabinets. At the end of a teaching day, spend just a few minutes writing notes in each of the lesson plans you have used that day. Think of these as notes to your future self—things that you want your future self to know about the next time the lesson plan is used.
Another aspect of personalization is organizing the material in a form that helps you to quickly find and retrieve a particular item you are interested in. If your personal DFC is on your own computer, you may want to use a structure of file folders. An elementary school teacher, for example, might want to have a separate file folder for each of the subject areas she or he teaches.
A DFC can be searched electronically, and it can have an indexing system that is specifically designed to fit the needs of its owner. This is a key idea. Think in terms of deciding upon a list of Categories (using the term the way a Wiki does), being able to easily add or delete categories, and being able to index the various entries in one's DFC. The combination of indexing using Categories and use of a search engine is a powerful aid to finding what is in your DFC.
A Sample Collection of Relevant Materials
Math is considered to be one of the basics of education. We want all students to gain contemporary levels of knowledge and skills in reading, writing, and math.
One of the specific goals of the IAE-pedia is to aid in the creation, collection, and dissemination of a Math Education Digital Filing Cabinet of materials designed to serve the needs of teachers and their students. The underlying model for this Digital Filing Cabinet is a collection of free, open source materials that can made available to teachers and students throughout the world.
The IAE-pedia contains a number of articles designed to help improve informal and formal math education. Some of these articles are listed below. The list given below also contains some links to articles located elsewhere.
The notion of "concrete," from concrete manipulatives to pedagogical sequences such as "concrete to abstract," is embedded in educational theories, research, and practice, especially in mathematics education. In this article, I consider research on the use of manipulatives and offer a critique of common perspectives on the notions of concrete manipulatives and concrete ideas. I offer a reformulation of the definition of "concrete" as used in psychology and education and provide illustrations of how, accepting that reformulation, computer manipulatives may be pedagogically efficacious.
Computational Thinking. Cuts across all disciplines. Includes an emphasis on math modeling that makes use of human brain and computers.
David Moursund Editorials. A collection of all of the editorials that David Moursund wrote for the Oregon Computing Teacher, The Computing Teacher, and Learning and Leading with Technology.
Empowering Learners and Teachers. This document includes a specific discussion of empowering students through teaching of reading and math. It includes the calculator and the digital watch in its examples.
Folk Math. A seminal article by Eugene Maier that draws a parallel between Folk Music (music that the ordinary people learn and do or use), and Folk Math. Contrasts Folk Math with School Math. See also: Eugene Maier. Gene is a world class math educator. This collection of short articles is well suited for use in preservice and inservice math education, and by others interested in the quality of math education that children are currently receiving.
Free Math Software. There is a huge and growing amount of free math software, math education software, math-oriented games, and so on.
Johnson, Jerry (n.d.). Math NEXUS. Jerry Johnson's Math NEXUS Website is an excellent example of a math education digital filing cabinet. It is designed to meet the needs of students and faculty interested in math education, and it serves as an outlet for his own personal creativity.
Lockhart, Paul (2002). A Mathematician's Lament. Retrieved 4/24/08: Argues that math should be considered an art, compares with music and painting, and "blasts" our current math education system.
Math Education. Discusses different answers to the questions, "What is mathematics." Emphasizes the need for students to gain increasing insight into possible answers as they progress in their math studies.
The Math Forum Is...the leading online resource for improving math learning, teaching, and communication since 1992.
We are teachers, mathematicians, researchers, students, and parents using the power of the Web to learn math and improve math education.
We offer a wealth of problems and puzzles; online mentoring; research; team problem solving; collaborations; and professional development. Students have fun and learn a lot. Educators share ideas and acquire new skills.
Math Maturity. An introduction to a general measure of student progress toward learning mathematics for long term use and understanding.
Introduction to Using Games in Education: A Guide for Teachers and Parents.
The Mind and the Computer: Problem Solving in the Information Age.
College Student's Guide to Computers in Education.
Moursund Editorial: High Tech—High Touch. Explores the need for education to provide an appropriate balance between high technology and strongly people-oriented low or no technology. From the November 1985 issue of The Computing Teacher.
International and domestic comparisons show that American students have not been succeeding in the mathematical part of their education at anything like a level expected of an international leader. Particularly disturbing is the consistency of findings that American students achieve in mathematics at a mediocre level by comparison to peers worldwide. On our own "National Report Card"—the National Assessment of Educational Progress (NAEP)—there are positive trends of scores at Grades 4 and 8, which have just reached historic highs. This is a sign of significant progress. Yet other results from NAEP are less positive: 32% of our students are at or above the "proficient" level in Grade 8, but only 23% are proficient at Grade 12. Consistent with these findings is the vast and growing demand for remedial mathematics education among arriving students in four-year colleges and community colleges across the nation.
Moreover, there are large, persistent disparities in mathematics achievement related to race and income—disparities that are not only devastating for individuals and families but also project poorly for the nation's future, given the youthfulness and high growth rates of the largest minority populations.
Oregon_Mathematics-OCTM. Email messages facilitating an increased level of communication and discussion among members of the Oregon Council of Teachers of Mathematics.
Problem Solving. Problem solving lies at the core of each academic discipline. Many of the general ideas and strategies used in problem solving in one discipline can transfer to other disciplines. This is especially true of math problem solving, since math is an important component of many other disciplines.
Science & Technology Museum Math Exhibit. Explores possible answers to the question, "What is math?" Analyzed some components of a science and technology exhibit on math. Explores the idea of making an elementary school classroom and overall curriculum more mathematical.
Two Brains Are Better Than One. Explores educational implications of human brain and computer brain working together to solve problems in math and other areas.
What is Computer Science? Computer science is a discipline closely related to mathematics. In many cases, today's Computer and Information Science Departments were "spun off" from Math departments. In many other cases, Computer Science and Math are still together in one college or university department.
MathWorld is the web's most extensive mathematical resource, provided as a free service to the world's mathematics and internet communities as part of a commitment to education and educational outreach by Wolfram Research, makers of Mathematica.
MathWorld has been assembled over more than a decade by Eric W. Weisstein with assistance from thousands of contributors. Since its contents first appeared online in 1995, MathWorld has emerged as a nexus of mathematical information in both the mathematics and educational communities. It not only reaches millions of readers from all continents of the globe, but also serves as a clearinghouse for new mathematical discoveries that are routinely contributed by researchers. Its entries are extensively referenced in journals and books spanning all educational levels, including those read by researchers, elementary school students and teachers, engineers, and hobbyists.
Women and ICT. ICT and mathematics overlap. Many of the women who were pioneers in the computer field were mathematicians.
The Math Forum is a leading center for mathematics and mathematics education on the Internet. Operating under Drexel's School of Education, our mission is to provide resources, materials, activities, person-to-person interactions, and educational products and services that enrich and support teaching and learning in an increasingly technological world.
Our online community includes teachers, students, researchers, parents, educators, and citizens at all levels who have an interest in math and math education.
Making math-related web resources more accessible
Want to use or develop educational technology? Visit Math Tools, the Forum's community digital library supporting the use and development of software for mathematics education. When a generic Web directory falls short of your mathematics needs, visit the Forum Internet Mathematics Library, which covers math and math education Web sites in depth. In our collaboration with the Mathematical Association of America, Mathematical Sciences Digital Library (MathDL), we collect mathematics instructional material with authors' statements and reader reviews; and catalogs mathematics commercial products, complete with editorial reviews, reader ratings and discussion groups. The Problems Library offers a convenient interface for searching and browsing the collective archives of the six Problem of the Week services.
Providing high-quality math and math education content
There's a lot of material on the Web, but how good is it, and how does it take advantage of new technologies or implement new pedagogy? We have worked with teachers, students, and researchers to put the best of their materials on the Web. This collaborative work is available via the Forum's Teacher Exchange: Forum Web Units. Teachers are invited to use the Web interface to contribute their own lessons.
This is an excellent and growing set of math materials for preservice and inservice math teachers at all grade levels. Some examples of the categories of material being made available include:
Problem of the Week.
Quote of the Week.
Statistic of the Week
Humor of the Week
Website of the Week
Resource of the Week
Also from the Math Forum, see their Library. This is a very large collection of materials and links to materials.
Math Resources from the Southern Oregon Education Service District. Retrieved 2/5/08: A nice collection of computer-based resources of use to teachers and to teachers of teachers.
Permission is granted to copy, distribute and/or modify these documentsTechnological Pedagogical Content Knowledge (TPCK) attempts to capture some of the essential qualities of knowledge required by teachers for technology integration in their teaching, while addressing the complex, multifaceted and situated nature of teacher knowledge. At the heart of the TPCK framework, is the complex interplay of three primary forms of knowledge: Content (CK), Pedagogy (PK), and Technology (TK). … the TPCK framework builds on Shulman's idea of Pedagogical Content Knowledge.
Research in the area of educational technology has often been critiqued for a lack of theoretical grounding. In this article we propose a conceptual framework for educational technology by building on Shulman's formulation of ''pedagogical content knowledge'' and extend it to the phenomenon of teachers integrating technology into their pedagogy. This framework is the result of 5 years of work on a program of research focused on teacher professional development and faculty development in higher education. It attempts to capture some of the essential qualities of teacher knowledge required for technology integration in teaching, while addressing the complex, multifaceted, and situated nature of this knowledge. We argue, briefly, that thoughtful pedagogical uses of technology require the development of a complex, situated form of knowledge that we call Technological Pedagogical Content Knowledge (TPCK). In doing so, we posit the complex roles of, and interplay among, three main components of learning environments: content, pedagogy, and technology. We argue that this model has much to offer to discussions of technology integration at multiple levels: theoretical, pedagogical, and methodological. In this article, we describe the theory behind our framework, provide examples of our teaching approach based upon the framework, and illustrate the methodological contributions that have resulted from this work.
The National Library of Virtual Manipulatives (NLVM) is an NSF supported project that began in 1999 to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-12 emphasis). The project includes dissemination and extensive internal and external evaluation.
Learning and understanding mathematics, at every level, requires student engagement. Mathematics is not, as has been said, a spectator sport. Too much of current instruction fails to actively involve students. One way to address the problem is through the use of manipulatives, physical objects that help students visualize relationships and applications. We can now use computers to create virtual learning environments to address the same goals.
There is a need for good computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels. Our Utah State University team is building Java-based mathematical tools and editors that allow us to create exciting new approaches to interactive mathematical instruction. The use of Java as a programming language provides platform independence and web-based accessibility.
The NLVM is a resource from which teachers may freely draw to enrich their mathematics classrooms. The materials are also of importance for the mathematical training of both in-service and pre-service teachers.
Lessons: "
Site Reviews: -- Tired of looking through page after page of search-engine hits trying to find a site that might have something useful? These categorized Internet links have all been reviewed by Purplemath.
Free Online Tutoring and Lessons
Quizzes and Worksheets
Other Useful Sites and Services
Only those sites with something immediately useful (and free) for algebra students are listed. You won't find math jokes, biographies, or recreational math sites here. Instead, check these review for sites containing lessons, tutoring forums, worksheets, articles on "how math is used in real life", and more.
Homework Guidelines: "How to suck up to your teacher." -- English teachers tell students explicitly how to format their papers. Math teachers, on the other hand, frequently just complain about how messy their students' work is. Neat homework can aid your comprehension and maybe make your teacher like you better. These Homework Guidelines for Mathematics will give you a leg up, explaining in clear terms what your math teacher is looking for.
Study Skills Self-Survey: "Do I have what it takes?" -- Much of your success or failure in algebra can be laid at the feet of your study habits. Do you have good math study habits? Take this survey and find out.
This five page research article includes a discussion of math learning. It begins by noting that most of what one learns in a course is not retained very long. It argues that a change in study habits can make a very large difference in long term retention.
Roughly speaking, the authors argue that the design of the seat work and homework in the typical math book is poor if one's goal is long term retention. In the two paragraphs that follow, Spacers divide their study time into two sessions with a space in between. Massers mass their study time into one concentrated session. Quoting from the article:
Because the experiments described thus far required subjects to learn concrete facts, it is natural to wonder whether the results of these studies will generalize to tasks requiring more abstract kinds of learning. To begin to explore this question, we have been assessing the effects of overlearning and spacing in mathematics learning. For example, in one experiment (Rohrer & Taylor, 2006), students were taught a permutation task and then assigned either three or nine practice problems. The additional six problems, which ensured heavy overlearning, had no detectable effect on test scores after one or four weeks. In another experiment with the same task (Rohrer & Taylor, in press), a group of Spacers divided four practice problems across two sessions separated by one week, whereas a group of Massers worked the same four problems in one session. When tested one week later, the Spacers outscored the Massers (74% vs. 49%). Furthermore, the Massers did not reliably outscore a group of so-called Light Massers who worked only half as many problems as the Massers (49% vs. 46%).
This apparent ineffectiveness of overlearning and massing is troubling because these two strategies are fostered by most mathematics textbooks. In these texts, each set of practice problems consists almost entirely of problems relating solely to the immediately preceding material. The concentration of all similar problems into the same practice set constitutes massing, and the sheer number of similar problems within each practice set guarantees overlearning. Alternatively, mathematics textbooks could easily adopt a format that engenders spacing. With this shuffled format, practice problems relating to a given lesson would be distributed throughout the remainder of the textbook. For example, a lesson on parabolas would be followed by a practice set with the usual number of problems, but only a few of these problems would relate to parabolas. Other parabola problems would be distributed throughout the remaining practice sets.
SAGE is a free open source alternative to Magma, Maple, Mathematica, and Mathlab. It is available for Windows, Mac OSX, and Linux. Quoting from the Website:
Use SAGE for studying a huge range of mathematics, including algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, and exact linear algebra.
SAGE makes it easy for you to use most mathematics software together. SAGE includes interfaces to Magma, Maple, Mathematica, MATLAB, and MuPAD, and the free programs Axiom, GAP, GP/PARI, Macaulay2, Maxima, Octave, and Singular.
The Shodor Foundation is a non-profit research and education organization dedicated to the advancement of science and math education, specifically through the use of modeling and simulation technologies.
Welcome to the Shodor's Curriculum Materials portal. There are several ways for you to browse our resources:
Search by grade level
Find the materials that are specifically geared towards a particular educational level.
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Locate all of the available Shodor resources for a subject or field of study.
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View a list of all of Shodor's Curriculum Materials projects. Use this tool to find a specific project that you have used before.
Interactive Teaching Environments
The Shodor Foundation staff and associates are developing interactive tools and simulations that enable and encourage exploration and discovery through observation, conjecture, and modeling activities. These Modeling and Simulation Technology for Education Reform (MASTER) tools are part of on-going collaborations with the National Center for Supercomputing Applications (NCSA) and other education organizations. Simulations and supporting materials developed by Foundation staff form the basis of international science collaborations presently demonstrating network technologies involving middle and high schools of the Department of Defense Education Activity (DoDEA).
A growing portfolio of MASTER tools are being fully integrated with new collaboration tools and on-line research facilities to create authentic scientific experiences. All tools, simulations, and supporting curriculum materials are designed in accordance with the National Science Education Standards and the National Math Education Standards.
Links to Other IAE Resources
This is a collection of IAE publications related to the IAE document you are currently reading. It is not updated very often, so important recent IAE documents may be missing from the list.
This component of the IAE-pedia documents is a work in progress. If there are few entries in the next four subsections, that is because the links have not yet been added. |
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Since arithmetic and geometric series are common menu items that everyone loves, they show up a lot in word problems. Unlike hotdogs and burgers, the word problems usually provide some individual quantities. They ask you to compute a related...
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Algebra And Trigonometry - 2nd edition
ISBN13:978-0201525106 ISBN10: 0201525100 This edition has also been released as: ISBN13: 978-0201640717 ISBN10: 0201640716
Summary: Interested in promoting interaction in the classroom? Then take a look at the Bittinger/ Beecher, Algebra and Trigonometry, Second Edition. This paperback text is designed specifically to motivate your students to participate actively and immedately. This paperback text is crafted to meet the varied skill level of your students-giving them the solid content coverage they need in a supportive format. Plus, these texts foster conceptual thinking with thinking and writi...show moreng exercises, computer/graphing calculator exercises, and a thoroughly integrated five-step problem solving approach. This worktext features a right triangle introduction to trigonometry. ...show less
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Elementary Statistics - Updated Solution Manual - 10th edition
Summary: Addison-Wesley is proud to celebrate the Tenth Edition of Elementary Statistics. This text is highly regarded because of its engaging and understandable introduction to statistics. The author's commitment to providing student-friendly guidance through the material and giving students opportunities to apply their newly learned skills in a real-world context has made Elementary Statistics the #1 best-seller in the market. Students learning from Elementary Statisti...show morecs should have completed an elementary algebra course. Although formulas and formal procedures can be found throughout the text, the emphasis is on development of statistical literacy and critical thinking. ...show less
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Fluid mechanics is an essential subject taught at degree level on engineering and science courses.
The book is complimentary follow up for the book "Engineering Fluid mechanics" also published on BOOKBOON, presenting the solutions to tutorial problems, to help students check if their solution method is correct, and if not, they can see the full solution hence giving them further practice in...
Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. The problems have a wide variation in difficulty, which is indicated by the number of stars.
This is an HTML version of the ebook and may not be properly formatted. Please view the PDF version for the original work.
An excerpt is a selected passage of a larger piece, hence this is not the complete book. |
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One of the most commonly asked questions in a mathematics classroom is, "Will I ever use this stuff in real life?" Some teachers can give a good, convincing answer; others hem and haw and stare at the floor. The real response to the question should be, "Yes, you will, because algebra gives you power" the power to help your children with their math homework, the power to manage your finances, the power to be successful in your career (especially if you have to manage the company budget). The list goes on.
Algebra is a system of mathematical symbols and rules that are universally understood, no matter what the spoken language. Algebra provides a clear, methodical process that can be followed from beginning to end to solve complex problems. There's no doubt that algebra can be easy to some while extremely challenging to others. For those of you who are challenged by working with numbers, Algebra I For Dummies can provide the help you need.
This easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems. But rest assured, this book is not about memorizing a bunch of meaningless steps; you find out the whys behind algebra to increase your understanding of how algebra works.
In Algebra I For Dummies, you'll discover the following topics and more:
All about numbers rational and irrational, variables, and positive and negative
Figuring out fractions and decimals
Explaining exponents and radicals
Solving linear and quadratic equations
Understanding formulas and solving story problems
Having fun with graphs
Top Ten lists on common algebraic errors, factoring tips, and divisibility rules.
No matter if you're 16 years old or 60 years old; no matter if you're learning algebra for the first time or need a quick refresher course; no matter if you're cramming for an algebra test, helping your kid with his or her homework, or coming up with next year's company budget, Algebra I For Dummies can give you the tools you need to succeed [via]
Volume I of a pair of classic texts and standard references for a generation this book is the work of an expert algebraist who taught at Yale for two decades. Volume I covers all undergraduate topics, including groups, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. 1985 edition.
An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence. [via]
. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics.
Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."
Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantum chemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.
This text provides a supportive environment to help students successfully learn the content of a standard algebra course. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, focus their studying habits, and obtain greater mathematical success. [via]
This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included. [via]
More editions of Commutative Algebra with a View Toward Algebraic Geometry:
This textbook for advanced courses in group theory focuses on finite groups, with emphasis on the idea of group actions. Early chapters identify important themes and establish the notation used throughout the book, and subsequent chapters explore the normal and arithmetical structures of groups as well as applications. Includes 679 exercises. 1978 edition.
Presents the fundamentals of linear algebra in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. This substantial revision includes greater focus on relationships between concepts, smoother transition to abstraction, early exposure to linear transformations and eigenvalues, more emphasize on visualization, new material on least squares and QR-decomposition and a greater number of proofs. Exercise sets begin with routine drill problems, progress to problems with more substance and conclude with theoretical problems. [via]
Noted for its expository style and clarity of presentation this substantial revision reflects a new generation of students' changing needs. Proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Features a wide variety of interesting contemporary applications which have been extensively revised and updated. Includes new material on least squares and QR-decomposition and greater emphasis on visualization. [via]
From the reviews: "The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity....The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher." --ZENTRALBLATT FÜR MATHEMATIK [via]
Considered a classic by many, A First Course in Abstract Algebra is an in-depth, introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. The sixth edition of this text continues the tradition of teaching in a classical manner while integrating field theory and a revised Chapter Zero. New exercises were written, and previous exercises were revised and modified. [via]
Galois theory is a fascinating mixture of classical and modern mathematics, and in fact provided much of the seed from which abstract algebra has grown. It is a showpiece of mathematical unification and of "technology transfer" to a range of modern applications.
Galois Theory, Second Edition is a revision of a well-established and popular text. The author's treatment is rigorous, but motivated by discussion and examples. He further lightens the study with entertaining historical notes - including a detailed description of Évariste Galois' turbulent life. The application of the Galois group to the quintic equation stands as a central theme of the book. Other topics include the problems of trisecting the angle, duplicating the cube, squaring the circle, solving cubic and quartic equations, and the construction of regular polygons
For this edition, the author added an introductory overview, a chapter on the calculation of Galois groups, further clarification of proofs, extra motivating examples, and modified exercises. Photographs from Galois' manuscripts and other illustrations enhance the engaging historical context offered in the first edition.
Written in a lively, highly readable style while sacrificing nothing to mathematical rigor, Galois Theory remains accessible to intermediate undergraduate students and an outstanding introduction to some of the intriguing concepts of abstract algebra. [via]
The Study Guide is based on David Lay's many years in the classroom, and has been updated so students can take full advantage of the new projects and data in the Updated Second Edition of the text. This guide gives the worked-out solutions to model problems that correspond with exercises in the text, along with study tips, hints to students, instructions for using MATLAB along with the text, additional MATLAB exercises, and expanded coverage of some text material. Maple and Mathematica appendices have been added, and the TI appendix has been updated to include coverage of the TI-86. [via]
Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a "brick wall.". Finally, when discussed in the abstract, these concepts are more accessible. Students' conceptual understanding is reinforced through True/False questions, practice problems, and the use of technology. David Lay changed the face of linear algebra with the execution of this philosophy, and continues his quest to improve the way linear algebra is taught with the new Updated Second Edition. With this update, he builds on this philosophy through increased visualization in the text, vastly enhanced technology support, and an extensive instructor support package. He has added additional figures to the text to help students visualize abstract concepts at key points in the course. A new dedicated CD and Website further enhance the course materials by providing additional support to help students gain command of difficult concepts. The CD, included in the back of the book, contains a wealth of new materials, with a registration coupon allowing access to a password-protected Website. These new materials are tied directly to the text, providing a comprehensive package for teaching and learning linear algebra. [via]
This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text. [via]
A student-oriented approach to linear algebra, now in its Second Edition
This introductory-level linear algebra text is for students who require a clear understanding of key algebraic concepts and their applications in such fields as science, engineering, and computer science. The text utilizes a parallel structure that introduces abstract concepts such as linear transformations, eigenvalues, vector spaces, and orthogonality in tandem with computational skills, thereby demonstrating clear and immediate relations between theory and application.
Introducing finite-dimensional representations of Lie groups and Lie algebras, this example-oriented book works from representation theory of finite groups, through Lie groups and Lie algrbras to the finite dimensional representations of the classical groups. [via] |
This chart maps out all the courses in the discipline and shows the links between courses and the minimum requirements for them. It does not attempt to depict all possible movements from course to course.
Note: MHF4U1 - While MCV4U can be taken concurrently with MHF4U, it is strongly recommended students take MHF4U before MCV4U
MPM1D1 (Academic) Principles of Mathematics This course enables students to develop understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a relationship. They will also explore relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multistep problems. Prerequisites: None Credits: 1 Note: Students considering Univerity programs, who earned mostly Level 3 and 4 in Grade 8 Math, and who are comfortable with abstract thinking in Math (e.g. Algebra), should consider this course.
MFM1P1 (Applied) Foundations of Mathematics This course enables students to develop understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will investigate real-life examples to develop various representations of linear relationships, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisites: None Credits: 1 Note: Students who learn best through a hands-on approach, and who struggled with some concepts in Grade 8 Math should consider this course which leads to College and some Univserity Level senior Courses. MAT1L1 (Locally Developed) Grade 9 Mathematics This course emphasizes further development of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, in the Grade 10 MAT2L1 (Locally Developed) course, and in the Mathematics Grade 11 and Grade 12 Workplace Preparation courses. The course is organized in three strands related to money sense, measurement, and proportional reasoning. In all strands, the focus is on developing and consolidating None Credits: 1 Note: Students who worked on Math expectations below Grade Level in Grade 8 should consider this class to have the opportunity to consolidate skills, and if successful could then take MFM1P1
MPM2D1 (Academic) Principles of Mathematics This course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relationships and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically as they solve multistep problems and communicate their thinking. Prerequisites: MPM1D1 Credits: 1
MFM2P1 (Applied) Foundations of Mathematics This course enables students to consolidate their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and hands-on activities. Students will develop and graph equations in analytic geometry; solve and apply linear systems, using real-life examples; and explore and interpret graphs of quadratic relationships. Students will investigate similar triangles, the trigonometry of right-angled triangles, and the measurement of three-dimensional objects. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisites: MPM1D1 or MFM1P1 Credits: 1 Note: Students who learn best through a hands-on approach, benefit from multiple representations or who struggled in the academic path should consider this course which leads to College and some University level senior courses.
MAT2L1 (Locally Developed) Grade 10 Mathematics This course emphasizes the extension of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, and in the Mathematics Grade 11 and Grade 12 Workplace Preparation courses. The course is organized in three strands related to money sense, measurement, and proportional reasoning. In all strands, the focus is on strengthening and extending MAT1L1, MFM1P1 or MPM1D1 Credits: 1 Note: This course is appropriate for students who need to consolidate skills after completing MAT1L1; students who achieve considerable success in MAT1L1 could consider MEL3E1. MCR3U1 (University) Functions This course introduces the mathematical concept of the function by extending students'ominal and rational expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisites: MPM2D1 Credits: 1 Note: Students who earn Level 3 or 4 with good Learning Skills in MPM2D1 are prepared for this course; students who earn Level 1 or 2 in MPM2D1 should consider the MCF3M1. MCF3M1 (University/College) Functions and Applications This course introduces basic features of the function by extending students' experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling Prerequisites: MPM2D1 or MFM2P1 Credits: 1 Note: MCF3M1 is a prerequisite for MCT4C1 and MDM4U1 courses; students who do very well in MFM2P1 (strong Level 3 and Level 4) or who need consolidation after MPM2D1 (earned Level 1 or Level 2) should consider this course.
MBF3C1 (College) Foundations for College Mathematics This analysing, and evaluating data involving one and two variables. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisites: MFM2P1 Credits: 1 Note: MBF3C1 is the prerequisite for MAP4C1, but MCF3M1 is required for MCT4C1 - study the flowchart at the beginning of this section to ensure you are choosing the right course for your post-secondary plans.
MEL3E1 (Workplace) Mathematics for Work and Everyday Life This Prerequisites: MPM1D1, MFM1P1, MAT2L1 or MFM2P1 Credits: 1 Note: MEL3E1 is appropriate for students who are planning to enter the work force or going on to College preparation programs after graduating. MCV4U1 (University) Calculus and Vectors This course builds on students previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors, and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, rational, exponential, and sinusoidal functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who plan to study mathematics in university and who may choose to pursue careers in fields such as physics, engineering and science. Prerequisites: MHF4U1 Credits: 1 Note: While MCV4U1 can be taken concurrently with MHF4U1, it is strongly recommended that students take MHF4U1 before MCV4U1.
MHF4U1 (University) Advanced Functions
This course extends students experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students who plan to study mathematics in university and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.
Prerequisites: MCR3U1 or MCT4C1 Credits: 1 Note: MHF4U1 is appropriate for students who showed considerable proficiency in MCR3U1 and who have good learning skills.
MDM4U1 (University) Mathematics of Data Management This course broadens students understanding of mathematics as it relates to managing data. Students will apply methods for organizing large amounts of information; solve problems involving probability and statistics; and carry out a culminating project that integrates statistical concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. Students planning to enter university programs in business, the social sciences, and the humanities will find this course of particular interest. Prerequisites: MCF3M1 or MCR3U1 Credits: 1
MCT4C1 (College)Offered 2014 - 2015 Mathematics for College Technology This course enables students to extend their knowledge of functions. Students will investigate and apply properties of polynomial, exponential, and trigonometric functions; continue to represent functions numerically, graphically, and algebraically; develop facility in simplifying expressions and solving equations; and solve problems that address applications of algebra, trigonometry, vectors, and geometry. Students will reason mathematically and communicate their thinking as they solve multi-step problems. This course prepares students for a variety of college technology programs. Prerequisites: MCF3M1 or MCR3U1 Credits: 1 Note: MCT4C1 is appropriate for students who showed considerable proficiency in MCF3M1 and is recommended by a variety of College Technology programs; though not required, students who take MCT4C1 are more successful in College Technology programs. MAP4C1 (College) Foundations for College Mathematics This course enables students to broaden their understanding of real-world applications of mathematics. Students will analyse data using statistical methods; solve problems involving applications of geometry and trigonometry; simplify expressions; and solve equations. Students will reason mathematically and communicate their thinking as they solve multi-step problems. This course prepares students for college programs in areas such as business, health sciences, and human services, and for certain skilled trades. Prerequisites: MBF3C1 Credits: 1
MEL4E1 (Workplace) Mathematics for Work and Everyday Life This course enables students to broaden their understanding of mathematics as it is applied in the workplace and daily life. Students will investigate questions involving the use of statistics; apply the concept of probability to solve problems involving familiar situations; investigate accommodation costs and create household budgets; use proportional reasoning; estimate and measure; and apply geometric concepts to create designs. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisites: MEL3E1 Credits: 1 Note: MEL4E1 is appropriate for students who are planning to enter the work force or going on to College preparation programs after graduating. back to top |
Industry Applications of Maple 17
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Description
This webinar offers a quick and easy way to learn some of the fundamental concepts for using Maple. Learn the basic steps on how to compose, plot and solve various types of mathematical problems. This webinar will also demonstrate how to create professional looking documents using Maple, as well as the basic steps for using Maple packages.
During the session we'll explore some of the new tools and resources available in Maple 17. Maple 17 contains new features that support all aspects of your work, including a new advanced code editor; embedded video functionality; a new Signal Processing and Group Theory package, as well as improvements to existing packages; and more! |
Description: This course is an introduction to modern geometry of
high-dimensional convex bodies. We are going to study the distances
between convex bodies, sections and projections, approximation of
systems of functions etc. In these problems the precise structure of
a convex body is usually unknown, so the probabilistic methods will
play a crucial role. In particular, to find a section of a convex
body with certain nice properties, one can consider a random section
and show that the property is satisfied with positive probability.
This approach allows to prove the celebrated Dvoretzky theorem: any
high-dimensional convex body possesses a section of a large
dimension, which is close to an ellipsoid. The existence of an
ellipsoidal section is a surprising result even for simplest convex
bodies, such as cubes or cross-polytopes.
We shall consider the properties of random convex bodies, the
connection between the volume of a body and the structure of its
sections and different applications of convex geometry techniques to
Analysis. |
North Carolina Modules (NC1, NC2, NC3, NC4, NC5)
North Carolina has recently emerged as a leader with regard to state
curriculum and assessment standards. The North Carolina Standard Course
of Study, which dates back to 1898, is revised periodically to reflect
changing needs of students and society. The latest revision adopts a
philosophy of competency-based curriculum.
In the early 1990's, the N.C. Department of Public Instruction (NCDPI)
produced an outline of competency goals by grade for mathematics, as well as
specific objectives and assessment measures pertaining to these goals. The
NCDPI also developed hundreds of problems which serve as sample measures and
provide a foundation for classroom instruction and assessment.
The North Carolina Math Objectives database (NC1) contains 1749
of those problems and 260 accompanying diagrams. The material spans skills
and concepts from 6th grade through high school (including calculus) and
includes multiple-choice, free-response, and open-ended problems. Teachers
in North Carolina will immediately see the benefit of having the problems in
a computer database. Teachers from around the country will certainly appreciate the wide
range of topics and the high-quality sample items for regular instruction
and assessment.
The North Carolina Elementary Math database (NC2) contains 1316
problems and 925 pictures taken from the state's released "Testlets" for
grades 3 through 5. The problems are organized by grade, and then by topic.
Although most problems are multiple-choice, there are some open-ended
questions available for each grade level.
The North Carolina Algebra I database (NC3) contains 2126
original multiple-choice problems and 600 pictures for the Algebra I
End-of-Course test. The original problems were authored by Susan Fink, a
respected math teacher from Forsyth County Schools (Winston-Salem), and are
drawing rave reviews from teachers across the state. All problems are cross referenced to the 1992 NC goals and objectives,
as well as to the ojectives enacted in 1998.
The North Carolina Reading database (NC4) contains 929 problems
based on more than 100 reading passages. The database contains all problems from
the North Carolina Dept of Public Instruction's testlets for grades 3 through 8,
and is organized by grade.
Finally, the North Carolina Math Testlets database (NC5) contains
1713 problems and 635 accompanying diagrams. This database is a continuation of
the NC2 database, as it includes all problems from the state's testlets
for grades 6 through 8, and Algebra I. The problems are organized by grade and
then by topic, and each grade includes a section of open-ended questions.
The North Carolina modules are available in packages at reduced prices for
North Carolina schools. NC schools, please contact EducAide for special
pricing information. |
This is a course in the algebra of matrices and Euclidean spaces that emphasizes the concrete and geometric. Topics to be developed include: solving systems of linear equations; matrix addition, scalar multiplication,
and
multiplication, properties of invertible matrices; determinants; elements of the theory of abstract finite dimensional real vector spaces; dimension of vector spaces; and the rank of a matrix. These ideas are used to
develop
basic ideas of Euclidean geometry and to illustrate the behavior of linear systems. We conclude with a discussion of eigenvalues and the diagonalization of matrices. For a more conceptual treatment of linear algebra,
students
should enroll in MATH223.
MAJOR READINGS
To be announced.
EXAMINATIONS AND ASSIGNMENTS
Two midterm exams, homework assignments, final exam for most sections, various problem sets and occasional quizzes for some sections. Students will take midterm exams at 7:30 p.m. on Monday, October 10 and Wednesday,
Novmber 16.
ADDITIONAL REQUIREMENTS and/or COMMENTS
MATH121, 122 or the high school equivalent is strongly recommended as background, but not required. |
MichMATYC 2002 Presenters
Haitham Al
Khateeb, University of
Indianapolis
Title: Undergraduate
studentsí understanding of division of fractions
Strands: Teacher
Preparation† 1-hour session
Abstract:† This
research study was designed to assess undergraduate students' understanding of
division of fractions. A paper and pencil instrument was administered as
a pre- and posttest to 59 undergraduate students who major in elementary
education. Analysis by independent t test of written responses provided
by students on the pre- and posttests showed lack of understanding, even
post-instruction.
John
Dersch, Grand Rapids CC
Title: Ahhh, the good old days:
The First 250 Years of Mathematics in America.†† 1-hour session
Strands: History of Mathematics†
Abstract:† What was mathematics like in America in 1700? In 1800? What
topics were taught at the college level? In the lower grades? What
were the textbooks like? What was the instruction like? What kind
of research was being done? How thorough was teacher training? Come
and find out! †Representative examples
of 18th and 19th century textbooks will be available for your perusal.
Jim
Ham, Delta College †Title: An Emerging Assessment Program†† 1-hour session
Strands: Assessment
Abstract: Delta College has been working on its assessment program for ten
years. This emerging model includes assessment at the classroom, course
and program levels. The mathematics faculty are involved in assessment
projects at all three levels. The presenter will provide updates of these
projects. In addition, the presenter will share successes, failures, and
some positive unintended consequences that resulted from the department's
engagement in assessment activities. Come share your own college's assessment
successes and challenges.
Barbara Jur, Macomb CC
Title:† "The Lens of the Udjat
Eye"†††† 1-hour session
Strands: History of Mathematics, applications/enrichment, and
developmental mathematics.
Abstract: Egypt has produced some important and useful mathematics, both
practical and instructive. From the use of unit fractions which the Greeks used
to practical geometry to study texts, the mathematics of the Nile region is
still of interest today. New discoveries and speculations are still being made
about what the ancients knew about mathematics.
Doug Mace, Kirtland CC
Title: Introduction to the MathWorks Project†††
1-hour session
Strands:† Application/Enrichment
Abstract:† Mathworks is an
interdisciplinary collection of open-ended laboratories designed by community
college mathematics faculty. Results of the usage of these laboratories
at Kirtland Community College will be discussed along with an introduction to
selected laboratories.
Jeff Morford, Henry Ford Community College
Title: Conclusions of Henry Ford Community College's Developmental Education
Task Force
Strands: Developmental Mathematics
Abstract: HFCC spent the last year reviewing its developmental program
campus wide. Come find out some of the conclusions we reached and some
changes we are planning. Find out what resources- colleges, articles and
books- that we used in crafting our report. Finally share what is new in
developmental math on your campus.
Kathy Mowers, Owensboro CC, Owensboro, KY
Title: Online Elementary Algebra: Can it work? †† 1-hour session
Strands: Developmental Mathematics, Distance Learning
Abstract: This presentation will focus on the presenter's experiences
teaching elementary algebra online including successes, challenges, and format.
It will also include her thoughts on ways to web-enhance other mathematics
courses, students' comments and her impressions and experiences.
Chuck Nicewonder, Owens CC, Toledo, Ohio
Title: Humor in the Mathematics Classroom?*.But Seriously†† 1-hour session
Strands: Articulation Abstract: This presentation will explore humor as a
necessary and fun component of any math class. Math-related jokes and other
bits of humor relating to various levels of mathematics, as well as their use
in the classroom and their effect on student performance, will be presented and
discussed. There will be time for input and discussion by all those in
attendance.
Jeffrey A. Oaks, University of Indianapolis
Title: Algebra and Inheritance in 9th Century Baghdad† 1-hour session
Strands: History of Mathematics
Abstract: To properly assess a medieval mathematical text we need to consider
both its relationship to previous works in the same field, and to the social
setting in which the work was produced. I will be examining the _Algebra_
of al-Khwarizmi from these perspectives. This will allow us to see not
only what is innovative and what is not in his work, but why he took his
particular approach to the subject.
Ann Savonen, Monroe County Community College
Title: Less Lecture = More Fun
Strands: Teacher
Preparation† 1-hour session Abstract: Do your math students love listening to long
lectures with lots of abstract concepts, theory, and definitions? Do long
reading assignments with the same information get them even more excited? If
so, do not come to this session. This session will present the idea of a
curriculum which minimizes lecture and the reading of long boring
textbooks. Instead, it encourages discovery, interaction, discussion,
hands-on experience, and fun. And yes, they will learn too!
Randy Schwartz, Schoolcraft College
Title:† Making Historical Arab
Mathematics Come Alive††† 1-hour session
Strands: History of Mathematics, Multicultural Mathematics
Abstract: Our curricula have scarcely acknowledged the scientific contributions
of non-European people.† My slideshow
illustrates why the medieval Arab world soared in mathematics.† Iíll also share activities whereby students
in Finite Math, Statistics, Linear Algebra and Business Calculus can use these
techniques to solve problems in combinatorics, linear modeling, and
optimization.
Abstract: Learn how
to use and incorporate new and exciting TI-83 + applications into your current
Algebra classes.† You will learn how to
use Algebra I, Inequalities, Transformations, Finance, and Polynomial Root
Finder Applications to motivate students and teach algebra more effectively.
Gwen Terwilliger, University of Toledo
Title:† Trials and Tribulations of
Teaching Math via Distance Learning†††
2-hour workshop
Strands: Distance Learning
Abstract:† Distance Learning encompasses
a wide variety of methods from videos to interactive multi-media Internet
presentation.† What works for one course
and/or instructor does not necessarily mean it will work for the next course
and/or instructor Ė or even for that same course and instructor for the next
group of students.† Also, distance
learning as a means for students to be able to complete their college education
is an excellent tool.† But, that does
not mean that all students will be able to succeed in this type of learning
environment any more than all students have†
one learning style.†† A successful
distance learning course takes at least or more time than teaching in a
traditional classroom.† This means that
any instructor planning or currently teaching a distance-learning course needs
1. careful planning of course, 2. ongoing assessment of the course, the
presentation, the students, etc., 3. immediate evaluation after the course is
completed for needed changes, and 4. constant learning about distance
learning† from what is available, what
works (and does not work) for others, about the students taking the course,
etc. This presentation will discuss a variety of resources and ideas for
teaching math via the Internet. Participants will be able to access some of the
Internet sites.
Mario F. Triola, Dutchess CC
Title:† Issues in Teaching
Statistics†† †††1-hour session
Strands:† Probability/Statistics
Abstract:† Why divide by n-1 for
standard deviation?† Why not use mean
absolute deviation?† What features make
a statistics course effective?† Which
technology should be used?† Are projects
important?† Which topics can be
omitted?† These and other important
issues facing statistics teachers will be discussed.
Deborah Zopf and Anna Cox, Henry Ford CC, Kellogg
CC
Title:† Letís Talk: Conversations about
Math for Elementary Teachers Courses††
1-hour session
Strands:† Teacher Preparation;
Collaboration Learning/ Learning Communities
Abstract:† This session will be an
informal conversation focused on Mathematics for Elementary Teachers
courses.† Participants will be
encouraged to bring ideas that they have employed while teaching these courses.
Highlights from the AMATYC Summer Session on Teacher Preparation will be given. |
Other Materials
Description
Algebra 1 will weave together a variety of concepts, procedures, and processes in mathematics including basic algebra, geometry, statistics and probability. Students will develop the ability to explore and solve mathematical problems, think critically, work cooperatively with others, and communicate their ideas clearly as they work through these mathematical concepts |
Chokoloskee PrecalculusTextbooks tend to have different definitions of Algebra 2, Trigonometry and Precalculus. Therefore |
The Mathematical Theory of Tone Systems
The Mathematical Theory of Tone Systems patterns a unified theory defining the tone system in functional terms based on the principles and forms of uncertainty theory. This title uses geometrical nets and other measures to study all classes of used and theoretical tone systems, from Pythagorean tuning to superparticular pentatonics. Hundreds of examples of past and prevalent tone systems are featured. Topics include Fuzziness and Sonance, Wavelets and Nonspecificity, Pitch Granulation and Ambiguity, Equal Temperaments, Mean Tone Systems. Well Tempered Systems, Ptolemy Systems, and more. Appendices include extended lists of tone systems and a catalogue of historical organs with subsemitones.
The Comparative Study of Electoral Systems systematically deals with the question of the impact of institutions on political behaviour. It provides comparative data on the micro- and the macro-level ...
A survey of the state of knowledge about the dynamics and gravitational properties of cosmic strings treated in the idealized classical approximation as line singularities described by the Nambu-Goto ...
This is an introductory textbook on the geometrical theory of dynamical systems, fluid flows, and certain integrable systems. The subjects are interdisciplinary and extend from mathematics, mechanics ...
This volume is a new revised printing which includes additional comments and incorporates corrections of misprints found in the first printing. It supplements "Fundamentals of the Physical Theory of ... |
Business Math For Dummies
Synopsis
The essential desk reference for every business professional or student
This easy-to-understand resource explains complex mathematical concepts and formulas and offers clear examples of how they relate to real-world business situations. Featuring practical practice problems to help readers hone their skills, it covers such key topics as working with percents to calculate increases and decreases, using basic algebra to solve proportions, and using basic statistics to analyze raw data. Readers will also find solutions for finance and payroll applications, including reading financial statements, calculating wages and commissions, and strategic salary planning.
Business Math For Dummies
eBook Information
ISBN: 9780470397398 |
19
Total Time: 2h 50m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 12/02/2008
Last Updated At: 10/27/2011
Taught by Professor Edward Burger, these lessons wereProfessor Burger does an amazing job once again! He explains the concept of "combining like terms" extraordinarily well. The reason his teaching style is so successful is he explains "why" a student most perform an action, which makes the "how" make sense and therefore immensely easier to grasp. Not to mention he is funny and entertaining, which makes the lesson actually fun to learn. Professor Burger, where were you when I was in High School Algebra??? Great job!
I can never quite remember how to calculate slope once i have two different points on the line, and then, depending on what other information I have, I run into problems generating the equation for the line. This really helped me understand not only how to use the formula but also how to think about it and why it works (and thus, hopefully, I'll remember it from now on).
This was really great. We homeschool, and it has been a long time since I used this. It was perfect for us to watch with our lesson.
Below are the descriptions for each of the lessons included in the
series:
Beg Algebra: The FOIL Method
The FOIL method is used in multiplying polynomials when each one only has two terms, for example (x+y)*(x-y). FOIL stands for First Outside Inside Last and indicates the order in which to multiply the terms of the polynomial. Professor Burger then provides you with a method to double-check that you have the correct number of terms. Next, you can combine like terms to simplify the equation. Finally, Professor Burger teaches an easy method to discern when your answer can be simplified to the difference of two squares Multiplying Polynomials
Once you have learned to add and subtract polynomials by combining like terms, Professor Burger teaches you how to multiply polynomials. First, he covers how to multiply powers with the same base. Then, he will cover how to raise one power to another power [(x^a)^b]. Finally, Prof. Burger will show you how to multiply two binomials, making sure you distribute every term.
For adding and subtracting polynomials Adding and Subtracting Polynomials
In this lesson, we learn how to add and subtract polynomials. A very important tip to keep in mind is to simplify the polynomial by combining like terms (terms with the same variable raised to the same power). When adding or subtracting strings of polynomials, you are simply combining the individual factors. Remember when subtracting that you need to distribute the subtraction sign to every factor within the parentheses.
Learn about multiplying polynomials in another lesson in the Beginner Algebra series:
equations Applying the Rules of Exponents
Professor Burger introduces you to the rules of exponents, including a classic mistake made when multiplying two numbers of the same base with different exponents. You will learn that A^n * A^m = A^(n+m). Then Professor Burger will teach you the next rule, what to do when you multiply two numbers of different bases, raised to the same power (A^n * B^n = (AB)^n). The final rule of exponents teaches you what to do when you have a base raised to an exponent, with the entire expression raised to another power (or (A^n)^m for Consecutive Numbers
Professor Burger walks you through a word problem to find consecutive numbers. First, you will read the problem and then define a variable for the numbers you need to find. Using this variable, you will write an equation to solve for the variable. Then, you can replace this variable in the equation and determine the consecutive numbers Introduction to Inequalities
Professor Burger discusses solving inequalities for one variable. He begins with reminders about adding, subtracting, multiplying, and dividing with both positive and negative numbers and the effect on the inequality sign. Then he demonstrates solving for a variable within an inequality, using the inequality 2(x + 3) < 4x + 10. You will then review interval notation (covered in a previous lesson) and three different ways to write the answer specialty Equations
In this lesson, Professor Burger discusses solving problems with absolute values. Remember that the absolute value of a number includes both the positive and negative value of that number. This means that an equation involving an absolute value means that you will have to solve for two equations, one equal to a positive value, and one equal to the negative value with 2 Absolute Values
In this lesson, you will learn how to solve an equation that has two absolute values. When beginning any equation with an absolute value, remember that, by definition, the absolute value of a number has both a positive and negative answer. You will also go over how to work an equation with a fraction inside an absolute value.
For a refresher on equations with one absolute value, see this lesson Inequalities
Reminding us of its definition, Professor Burger demonstrates how to work an inequality with an absolute value. You will need to convert the inequality from the absolute value to an inequality encompassing both the positive and negative points of that absolute value. This will look different, depending on whether the absolute value is < or >. Prof. Burger walks you through several examples.
For an introduction to inequalities, see this lesson:
And for more on absolute values Factoring Trinomials Completely
In this lesson, you will learn how to factor trinomials using a reverse-FOIL trial-and-error method. You will start by simplifying the trinomial as much as possible, by removing any common factors or grouping any possible combinations. Then, try the reverse-FOIL by first breaking up the squared term. He also gives you a hint that when the last factor of the trinomial is negative, you know that the last terms of the binomials have to be opposites. You will walk through this process with a one-variable trinomial, a two-variable trinomial, and a trinomial with a positive last term expressions Direct Proportion
In order to explain direct proportionality, Professor Burger uses a real-world example of a spring and Hooke's Law. Hooke's law states that the distance a spring stretches varies directly to the force applied. If force, f, is directly proportional to distance, d, then d~f or d=kf. This equation allows us to find the constant, k, of how much the spring stretches when force is applied. After we have found this number, we can determine the distance the spring will stretch with varying forces applied.
A lesson on inverse proportions can be found here Inverse Proportion
In this lesson, inverse proportionality is explained using light as a real-world example. The illumination of a light source varies inversely to the square of the distance from the source, or I=k/(d^2). So, to find the illumination of a particular light source, you will need to find the constant, k, of that source, and then divide by the distance squared.
An introduction to direct proportion can be seen here:
the Slope Given Two Points
In this lesson, you will learn how to find the slope, or the relative increase (also known as a pitch), of a line if you are given two points on that line (x1, y1) and (x2, y2). The slope (denoted by the letter, m) of a line is defined by the change in y divided by the change in x. First, you must calculate the change in the two distances (or the change (x1 - x2) and the change (y1 - y2)). You will also learn the shorthand for writing the equation of a slope and the phrase 'rise over run.' After learning how to find the slope of a line, you will practice with several sets of points and lines with different slopes (including verticle lines and horizontal lines) and also practice graphing those lines to view the slope. Professor Burger also examines what it means when a slope is undefined (or the change in x = 0), and when a slope = 0 (or the change in y = 0 Distinguished in Slope-Intercept Form
Professor Burger teaches the algebraic expression for lines, or the equation of a line. The standard form for a line is written Ax + By = C. More complex algebraic expression include the slope-intercept form, y = mx + b, where m=slope and b= the point where the line crosses (or intercepts) the y-axis. Professor Burger proves the validity of this expression, and shows you how to graph a line from the slope-intercept equation.
Learn how to determine the slope of a line here equations Given Two Points
Using the slope-intercept equation of a line, Professor Burger teaches you how to write the equation of a line if you are given two points on that line. Given two points, you can find the slope. Once you have found the slope of the line, you can input any point on that line into the equation with the slope to solve for b. Once you have found the slope and b, you have the slope-intercept equation (y = mx + b).
Learn how to find the slope of a line Writing Point-Slope Form Equations
This lesson introduces another format for the equation of a line called the point-slope form. The point-slope form for the equation of a line is y - y1 = m(x - x1), where m=slope and x1 and y1 are the coordinates of a point on the line (x1, y1). Professor Burger proves the validity of this equation, which is derived from the formula for the slope.
Learn how to find the slope of a line in Parallel & Perpendicular Line Slopes
Professor Burger explains parallel and perpendicular lines, teaching you how to identify if two lines are parallel or perpendicular, by looking at the formulas. Two lines are parallel if they have the same slope. Perpendicular lines are slightly more complicated, as they have slopes that are negative reciprocals. After demonstrating these principles, Professor Burger walks you through some example problems.
To learn more about slopes Find distance & midpoint betwn points
Professor Burger proves the distance formula (distance = square root [(x1-x2)^2 + (y1-y2)^2] ) using the pythagorean theorum. Using this formula, you can find the distance between two points on a line. Professor Burger goes on to prove the midpoint formula ( [x1 + x2]/2, [y1 + y2]/2). The midpoint formula is in the form of a point on a line and is the average of the points on the x and y axes Quadratics by Factoring
In this lesson, you will learn to solve quadratic equations by factoring. Quadratic equations involve factors that are now squared, which could give us more than one possible answer. To discover if an equation has more than one answer, you need to set the equation equal to zero and factor. You will discover that if your quadratic equation factors into a perfect square, it will have only one solution |
Better student preparation needed for university maths: UK study
Aug 01, 2012
Moving from sixth form, or college, into higher education (HE) can be a challenge for many students, especially those who start mathematically demanding courses. Life prior to university focuses on achieving maximum examination success to be sure of a place. Faced with this pressure, school and college maths courses pay little attention to preparing students to use maths in other areas of study according to a project funded by the Economic and Social Research Council (ESRC).
A student's ability to apply mathematical reasoning is critical to their success, especially in HE courses like science, technology, engineering and medicine. The study, undertaken by Professor Julian Williams, Dr Pauline Davis, Dr Laura Black, Dr Birgit Pepin of the University of Manchester and Associate Professor Geoffrey Wake from the University of Nottingham, shows that it is important to understand how students can prepare for the 'shock to the system' they face and how they can be given support at school, college and university to help in the transition.
The researchers found that students were not fully aware of the importance of the mathematical content in the courses they had joined at university, and particularly how to apply maths in practice.
Associate Professor Geoffrey Wake states, "Different teaching styles of university lecturers and the need for autonomously-managed learning, where students need to learn some mathematical content of their courses on their own without input from lecturers, also came as a bit of a shock for many students. On the other hand, some of the lecturers had limited knowledge of the exam-driven priorities of A-level maths courses and were not aware of the techniques students had been taught prior to attending their university courses."
The researchers also found significant problems in motivating students to engage with the mathematics within their chosen university coursewhere mathematics was not their main area of study. Generally, schools and colleges were found not to be preparing students for university learning practices, and the level of learning-skills support was variable once students arrived at university.
"Many students felt that they would benefit from student-centred learning and greater opportunity for dialogue with their lecturers," says Associate Professor Wake. "Unfortunately, the efficiencies required of university teaching resulting in lecturing of large numbers of students makes developing such a learning culture unlikely."
The findings led the researchers to consider the implications for the policies and practices of schools, colleges and universities recommending a better two-way flow of information between schools and colleges and universities to address the issues of preparation and expectation.
They concluded that the sixth-form curriculum should provide 'learning to learn' skills and mathematical modelling for students following A-level maths courses.
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Module description
The module has three main aims:
to introduce the basic objects of mathematics (numbers, sets and
functions), and their properties;
to emphasize the fact that mathematics is concerned with proofs,
which establish results beyond doubt, and to show you how to construct
proofs, how to spot false "proofs", how to use definitions, etc.;
to get you involved in the excitement of doing mathematics.
The module description, with the syllabus and learning outcomes,
can be found here in the list of modules on the School's web page.
Here is an index to the notes, which might help you
to find your way around; and here are the study skills collected into a single document.
Problem sheets
The timing for problem sheets has been changed, starting from Sheet 4. The
dates on the sheet will be the week in which the material will be discussed
in tutorials; you can hand it in any time between the end of the tutorial and
the beginning of the following week's tutorial.
Administrative matters
Extra questions
A sample exam paper can be found here. Solutions
to the sample exam will not be published, but the lecturer is happy to go
through your attempts at this paper with you; please email for an appointment.
I have put the sample test and
test papers here. I have also posted
a list of statements made on test papers; you
are asked to explain what is wrong with each, and to try to clear up the
misunderstanding of the person who wrote it. |
Book DescriptionMore About the Author
Product Description
Review
Second Edition S. Lang Introduction to Linear Algebra "Excellent! Rigorous yet straightforward, all answers included!"—Dr. J. Adam, Old Dominion University
--This text refers to an out of print or unavailable edition of this title.
Inside This Book(Learn More)
Browse and search another edition of this book.
First Sentence
The concept of a vector is basic for the study of functions of several variables. Read the first page
This text is intended for a one semester introductory course in Linear Algebra at the sophomore level geared toward mathematics majors and motivated students. It was originally extracted from Lang "Linear Algebra," and is now in its second edition (a vast improvement over the first: Lang rarely does the increasingly popular token update).
The text takes a theoretical approach to the subject, and the only applications the reader can expect to see are to other interesting areas of mathematics. With the exception of the last chapter, these are left in the exercises, and Lang does not push them vary far.
The trend in most Linear Algebra texts at this level that attempt to appeal to a large audience (such as engineering students) is away from the Definition-Theorem-Proof approach and towards a less formal presentation based around ideas, discussions as proofs, and applications. I prefer the former approach, which Lang is very much in the tradition of, and believe that the way to teach students how to write rigorous and presentable proofs is by making them read and study them. In fact, I learned how to write proofs from studying this text and working all of Lang's well-chosen exercises.
"Introduction to Linear Algebra" starts at the basics with no prior assumptions on the material the reader knows (the Calculus is used only occasionally in the exercises): the first chapter is on points, vectors, and planes in the Euclidean space, R^n. After that is a chapter introducing matricies, inversion, systems of linear equations, and Gaussian elimination. While the book does spend adequate time on how to perform Gaussian elimination and matrix inversion, it also gives all the proofs that these methods work.
The bulk of the theoretical material comes in Chapters III through V, which respectively present the theories of vector spaces, linear mappings, and composite and inverse mappings. The approach is rigorous, but by no means inaccessible. As is necessary in a course like this, time is spent on establishing clear and solid proofs of basic results that will be treated as almost trivial ("you can show it on your homework to convince yourselves") in more advanced classes - c.f. Lang's "Undergraduate Algebra."
The next two chapters cover scalar products and determinants, and have a somewhat more computational feel to them. There is much theory in the sections on scalar products, but a big focus is also the Gramm-Schmidt method for finding an orthonormal basis. Many of the determinant proofs are in the 2 x 2 and 3 x 3 case to avoid bringing in the full formalism and notation of determinants in general.
The text concludes with what is its most difficult chapter, the one on eigenvectors and eigenvalues. It is the most, however, for applications to physics, and interest applications comprise the last half of the chapter.
If you are ordering this text used, I recommend you take care to find the second edition. The first edition was significantly shorter and covered less material.
This is an introductory text, and not for learning the material that would be included in a second course or part of the algebra sequence at the junior/senior level. For those purposes, I recommend Lang's "Linear Algebra." Portions bear strong (often exact) resemblance to the book at present consideration, but the most basic material is missing and much advanced material is included.
In conclusion, I highly recommend this text for a motivated student who wants a first exposure to Linear Algebra. The text isn't always easy reading, and parts may be a tough climb for readers without much exposure to this type of reading. The experience, however, is well worth it; in mathematics, one really only learns as much as one sweats, so to speak.
15 of 15 people found the following review helpful
5.0 out of 5 starsA wonderful book and benchmark test for students26 April 2000
By A Customer - Published on Amazon.com
Format:Hardcover
This is a wonderful book for freshmen/sophomores. Being a senior now, it's easier to evaluate the quality of the text and judge it's worth compared to other books. I really don't think there's a better book on linear algebra at this level. Everything in here is well motivated, organized and as rigorous as possible for an intro book. That's not to say that there's not room for improvement as far as motivation goes, but what he has certainly suffices. Even if you don't get everything in here on your first pass, this book provides a good benchmark test - if you can get through it in good shape, then you are probably well prepared to begin upper level work. If you can't, then you should probably try again before attempting a serious course in, say, group theory or topology. Linear algebra provides the ideal subject matter with which to introduce the student to rigorous proof techniques, because it has so many easily visualized yet useful examples. So if you can't follow the proofs here, don't expect to follow the proofs in a more abstract course. If there's any other book that I might use in this one's place, it would actually be Lang's "Linear Algebra," which I find to be more cohesive and motivated, although more difficult.
This text is well written and is motivated by theory. Better suited as a supplemental text, as opposed to a required course text. As for the self acclaimed "smart" reviewer from Irvine, A grades in mathematics do not mean that you have mastered the subject. The school which you attend also affects your grades. Not to mention, linear algebra is usually a bridge to higher mathematics, grades in calculus, and diff equs don't really matter. If you are having trouble with this text or the course associated with it, chances are that you will have a very, very hard time in more mathematical courses such as abstract algebra and classical analysis. Calculus 1, 2, 3, and diff equs are just applications of mathematical theory. It is doubtful that after this sequence that the student even knows the definition of a limit of a function in a single variable which is ironic, for what is calculus but the study of limits. For example, the derivative is the limit of the difference quotient, the Riemann integral is the limit of Riemann sums, etc...
The point is that linear algebra, at the theoretical level, is a bridge to higher mathematics. This is a good text to use in order to cross that bridge. Serge Lang is a great mathematician, he was recently given an award from the AMS for his achievements in writing textbooks. My favorite linear algebra text is 'Linear Algebra' by Hoffman/Kunze. For people who lack in mathematical ability and wish for a more applied introduction to linear algebra, 'Linear Algebra and its Applications' by Gilbert Strang. If the reader from Irvine or anyone having difficulty with this beautiful subject wishes for a even simpler text, there is 'Elementary Linear Algebra' by Bernard Kolman.
Amongst other great works by Serge Lang, I believe that 'Algebra' is a classic which should be in the library of any mathematics student and professor. |
Developing Skills in Algebra
Developing Skills in Algebra is designed for the
student who needs a comprehensive review of the topics from elementary and
intermediate algebra. This textbook uses the topics covered by many schools
in an intermediate algebra course. Within the reader friendly styled text,
students will find the algebraic skills necessary to prepare them for courses
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Elementary Number Theory;imal prerequisites make this text ideal for a first course in number theory. Written in a lively, engaging style by the author of popular mathematics books, it features nearly 1,000 imaginative exercises and problems. Solutions to many of the problems are included, and a teacher's guide is available. 1978 edition. Written in a lively, engaging style by the author of popular mathematics books, this volume features nearly 1,000 imaginative exercises and problems. Some solutions included. 1978 edition. |
Equations, Roots & Exponents Mastery DVD A 12-lesson pre-algebra program that teaches selected critical concepts, skills, and problem-solving strategies needed to recognize and work with different types of equations problems. |
In the Real World
In this section we left the integrating to the calculators. That's practical, because the calculators can integrate much more quickly and accurately than we can. Instead, we've been spending our time practicing imagination.
The word "imagination"...
Please purchase the full module to see the rest of this course
Purchase the Area, Volume, and Arc Length Pass and get full access to this Calculus chapter. No limits found here. |
Book Description: This 2008 book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of students and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others.
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Video Summary: This learning video is designed to develop critical thinking in students by encouraging them to work from basic principals to solve a puzzling mathematics problem that contains uncertainty. One class session of approximately 55 minutes is necessary for lesson completion. First-year simple algebra is all that is required for the lesson, and any high school student in a college-preparatory math class should be able to participate in this exercise. Materials for in-class activities include: a yard stick, a meter stick or a straight branch of a tree; a saw or equivalent to cut the stick; and a blackboard or equivalent. In this video lesson, during in-class sessions between video segments, students will learn among other things: 1) how to generate random numbers; 2) how to deal with probability; and 3) how to construct and draw portions of the X-Y plane that satisfy linear inequalities |
Elementary Computer Mathematics - Kenneth R. Koehler
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Elliptic Geometry Drawing Tools - Brad Findell
Elliptic geometry calculations using the disk model. Includes scripts for: Finding the point antipodal to a given point; drawing the circle with given center through a given point; measuring the elliptic angle described by three points; measuring the
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"Information Provider to the World." Elsevier's mission is "to advance science, technology and medical science by fulfilling, on a sound commercial basis, the communication needs specific to the international community of scientists, engineers and associated
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Embedded TEX - David McCabe
Mathematics typesetting for Word, Excel, other Microsoft Office programs, or any application that supports ActiveX. Download and install free trials. See also McCabe's tutorial, which explains the Fourier transform and Fourier series.Energy for Markov Chains [PDF] - Peter G. Doyle
A "very preliminary" paper touching on the Dirichlet principle, Escape probabilities and commuting time, The Monotonicity Law, and a Probabilistic interpretation of energy. PostScript and source file are available from Doyle's site.
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Engauge Digitizer - Mark Mitchell
Software that takes a digital graph or map, and converts it into numbers. Engauge Digitizer automatically traces curves of line plots and matches points of point plots; handles cartesian, polar, linear and logarithmic graphs; supports drag-and-drop and
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The Enigma Machine - Russell Schwager
This Java applet simulates the operation of an Enigma machine, based on the one used by the Germans in World War II to encrypt military messages. With information on the history and workings of the mechanical Enigma machine; a brief description of how
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Entropia.com
The Global Internet Community for Science and Mathematics Distributed Research Computing, Entropia.com is a community dedicated to performing real research on a massive scale using ordinary home and office computers connected through the Internet. Entropia.org's |
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Volume 6, Applications of Algebra teaches how to build equations to solve difficult story problems typically found on the SAT and College Placement Tests. Volume 5 teaches how to solve equations and Volume 6 teaches how to create the equations. $49 plus $10 shipping to any US State.
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DESCRIPTION;ENCODING=QUOTED-PRINTABLE:ABSTRACT: Although physics faculty are incorporating computers to enhance physics education, computation is often viewed as a black box whose inner workings need not be understood. We propose to open up the computational black box by providing Computational Physics (CP) curricula materials based on a problem-solving paradigm that can be incorporated into existing physics classes, or used in stand-alone CP classes. The curricula materials assume a computational science point of view, where understanding of the applied math and the CS is also important, and usually involve a compiled language in order for the students to get closer to the algorithms. The materials derive from a new CP eTextbook available from Compadre that includes video-based lectures, programs, applets, visualizations and animations.
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Geometry
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Summary: This is a systematic introduction to various geometries, including Euclidean, affine, projective, spherical, and hyperbolic. Also included is a chapter on infinite-dimensional generalisations of Euclidean and affine geometries.
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Looking at Learning ... Again, Part 2: Workshop 2. Mathematics: A Community Focus With Dr. Marta Civil. As teachers, we often make assumptions about the knowledge children are exposed to at home. Sometimes it seems that we focus on only reading and writing,Dr. Civil contends that we need to look more carefully at the mathematical potential of the home and that it is essential that schools learn to be more flexible and knowledgeable about students' home environments. See and hear from Dr. Civil, the teachers she works with, and a long-standing parent mathematics group, and fo Author(s): Harvard-Smithsonian Center for Astrophysics QSO top12.400 The Solar System (MIT) This is an introduction to the study of the solar system with emphasis on the latest spacecraft results. The subject covers basic principles rather than detailed mathematical and physical models. Topics include: an overview of the solar system, planetary orbits, rings, planetary formation, meteorites, asteroids, comets, planetary surfaces and cratering, planetary interiors, planetary atmospheres, and life in the solar system. Author(s): Binzel63 Advanced Fluid Dynamics of the Environment (MIT) Designed to familiarize students with theories and analytical tools useful for studying research literature, this course is a survey of fluid mechanical problems in the water environment. Because of the inherent nonlinearities in the governing equations, we shall emphasize the art of making analytical approximations not only for facilitating calculations but also for gaining deeper physical insight. The importance of scales will be discussed throughout the course in lectures and homeworks. Mathe Author(s): Mei, Chiang,Li, Guangda050 Engineering Mechanics I (MIT) This subject provides an introduction to the mechanics of materials and structures. You will be introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of materials and structures and you will learn how to solve a variety of problems of interest to civil and environmental engineers. While there will be a chance for you to put your mathematical skills obtained in 18.01, 18.02, and eventually 18.03 to use in this subject, the emphasis is Author(s): Ulm, Franz-Josef,B100A Analysis I (MIT) Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space.
MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications whe Author(s): Matt6.251J Introduction to Mathematical Programming (MIT) This course offers an introduction to optimization problems, algorithms, and their complexity, emphasizing basic methodologies and the underlying mathematical structures. The main topics covered include:
Theory and algorithms for linear programming
Network flow problems and algorithms
Introduction to integer programming and combinatorial problems Author(s): John Tsitsiklis
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Applied mathematics (bachelor)
Applied mathematics is not a particular branch of mathematics, but rather a way of working with mathematics. In the Applied mathematics programme, you will learn to create models of and solve problems from the practical world by utilising advanced mathematical tools. Read more about the programme
In the Applied mathematics programme, you will learn the basic principles of mathematics as well as be educated in mathematical modelling in order to perform analyses and calculations digitally. This way you will be capable of formulating complex problems which can then be solved. Read more about the programme structure
A Career in Applied Mathematics
An applied mathematician works with solving problems which lie beyond the realm of regular mathematics, and therefore Applied Mathematics is the programme for you if you want to work with mathematics and still keep in mind how mathematics can be of use in your future career. The Applied Mathematics programme provides a solid base for a career within the private business sector or a research facility, as well as within the field of education. |
A psychologist looks at why algebra is so stressful for so many students of all ages and what can be done to make it stress-free. If you or anyone in your family struggled with algebra in school, this article is for you.
Learning Algebra
on the Right Side of the Brain How to make algebra a successful learning experience
for students of all ages
James J. Asher, Ph.D.
Copyright 2001
Mathematicians are fond of saying that God must have been an algebraist (one versed in algebra). The reason: algebra impressed them as a divine code that God must have used to design the universe and all the natural events in that universe. For example, Einstein used a special algebra discovered by William Rowan Hamilton to predict the location of the planet Mercury---an extraordinary feat that made Albert an international celebrity in the world of science for an esoteric theory called relativity.
There is no quarrel that algebras are powerful languages (I used the plural because most people are not aware that there is more than one algebra). These algebras are only intellectual toys created in the minds of mathematicians. Some speculate that mathematicians have, in some clairvoyant way, been "reading" the divine mind. The algebraic models are completely independent of the external world. But, the bizarre fact is that these intellectual toys, somehow in some yet to be discovered way, give scientists permission to predict and often control natural events in chemistry, physics, and even medicine.
So why do students fail?
Why is algebra (perhaps the "language of God") such a high-stress, high-failure subject? One of every two students fails the course and walks away with the harmful conclusion, "I guess that I am no good at mathematics!" I want to present three explanations for this strange result that disables 50 percent of our student population. I will conclude with solutions that promise an exciting turn-around for students of all ages.
Explanation Number 1 for why students fail
There is an assumption that algebraic exercises, especially word problems, improves problem solving and thinking. There is no evidence that solving those word problems transfers to other intellectual skills such as problem solving, creativity, or thinking. Solving those word problems only makes one proficient at solving more word problems exactly like those in the textbook.
A closer look at word problems
Seventh-graders were asked to solve this word problem: "Orville and Wilbur owned a bicycle shop which also sold tricycles. One day, they decided to take an inventory of their stock. They each volunteered to count one item, which would have worked out just fine if one had counted bicycles and the other had counted tricycles. But Orville and Wilbur were both very independent thinkers. Orville counted the number of pedals in the shop and Wilbur counted the number of wheels.
"Orville found that they had 144 pedals in the shop, and
Wilbur found that they had 186 wheels. All pedals and wheels
were actually parts of either bicycles or tricycles. They
were just about to start over with their inventory when their
friend Kitty, who was a good problem solver, challenged them
to figure out the number of bicycles and tricycles from the
inventory they had already done. Can you help the Wright
brothers? How many bicycles and tricycles did they have in
their shop?..." (San Jose Mercury News, April 3, 1995).
Some kids perceive this as a fun puzzle and joyfully
speculated about possible ways to develop an answer. Other
youngsters perceive this word problem as absolute nonsense.
They reason: We are talking about the Wright brothers, owners
of a bicycle shop in Ohio. The brothers are famous for doing
the impossible---inventing a bicycle that flies in the air.
Secondly, these thoughtful students (who probably will get
"F" in algebra) do not believe that the geniuses who invented
the airplane would waste valuable hours counting wheels and
pedals when the simple solution is to count bicycles and
tricycles. Surely these intellectual giants have something
better to do with their time.
Nobody cares about word problems---
not even the writers of algebra textbooks
The reason we find nonsensical word problems in those textbooks is that it is impossible to find meaningful problems in real life. (I challenge anyone to e-mail me one meaningful word problem from the real life of ordinary people that can be solved with algebra. You can reach me at [email protected]) Writers must invent synthetic word problems that are of no interest to anyone, including those who wrote the textbook. For example, Ellen is 7 years older than her sister, and the sum of their ages is 21 years. How old is each? First, this is a puzzle and not a problem because no one cares one way or the other about Ellen or her sister.
Secondly, not only is the answer already known, but the answer came before the question. Unless you already know the ages of both Ellen and her sister, how can you conclude (a) that Ellen is 7 years older than her sister, and (b) the sum of their ages is 21 years? The ages of Ellen and her sister had to be known in advance. So why ask the question?
Here is another example taken from a textbook used in the first algebra course. Notice that the question will be of interest to no one---not even the author of the algebra text: "In a class of 37 pupils there are five more girls than boys. How many boys and how many girls are there?" We could continue with those mind-numbing word problems about trains going in opposite directions and meeting somewhere on the journey. Why would anyone want to predict where they will cross paths and how long it will take to do this? I wonder whether professional railroad personnel have ever bothered to sort out the answers to this puzzle. Does it have a trace of relevancy to the operation of a railroad? Is it of any interest to passengers on a train? If it is of no concern to those who operate or ride trains, why should it be of concern to us?
Explanation Number 2 for why students fail
There is the assumption that algebra is absolutely positively an essential skill for boys and girls in all walks of life. Not only does everyone need algebra, but students cannot hope to pass those entrance examinations to enter college without an understanding of algebra.
I invite you to visit any shopping mall in America, stop ten people at random, and ask this question: "Once you were out of school, can you think of a time in your life when you used algebra to solve an important problem? If so, what was it?"
I predict that you will not find one person in ten who will answer in the affirmative. Even airline pilots have tables and ready-made graphs for plotting distances and estimating time of arrival. People in finance have ready-made tables for finding the answers to financial problems that they frequently encounter such as compound interest. Most people are successful in their everyday lives without using algebra.
Well then, how about getting into college? We need algebra for that. True, but this is an artificial gate for admittance. It is like the requirement that candidates for officer training, especially flying, be a college graduate. In an interview with an Israeli Air Force general, 60 Minutes reporter, Mike Wallace, discovered that one did not have to be a college graduate to be accepted into the Israeli flight program---a training experience that produces excellent fighter pilots. "Why then," Wallace asked, "does the U.S. Air Force insist upon a college education before a person can enter our flight training?"
"Mike," the general responded, "I don't know. It may be one of those things we assume is necessary, but have no proof one way or the other."
Algebra is a screening device for college entrance much as Latin was a hundred years ago. How can anyone consider themselves educated without Latin? Proof that precollege algebraic skill is essential for success in college is non-existent.
Explanation Number 3 for why students fail
When students ask, "Why do I have to take algebra?" The
answer is quasi-religious: "Trust me! You will need algebra
to be successful as a scientist, engineer or doctor. You will
need algebra to take college chemistry, physics, and
mathematics. Algebra is a must!"
Merely asserting that algebra is valuable is not enough. This is like a car ride with children in the back seat who keep asking their parents, "Where are we going?" "When will we get there?" You will not quiet the children with, "We are on our way to Saint Louis. We will arrive in five days." The children will want to stop frequently before Saint Louis. Some of their favorite places are the colorful balls in the play area at MacDonalds, and the swimming pool at the motel.
In other words, the teacher's goal is not necessarily the student's goal. The children do not believe there is a place called Saint Louis. The students do not believe that a long mathematical journey with no attractive places along the way is worth the effort.
Well then, what do you recommend?
I recommend three options we can try. The first is to make algebra an elective rather than a mandatory course for all students. Many students enjoy the intricate pattern- making activity of algebra. These students find the patterns fascinating apart from any synthetic attempt to make the product relevant. They will enjoy the course. Algebra should be declassified from its current status as "something everyone has to know" to "here is another interesting elective you may enjoy along with art, botany, or sports."
But, what about those who "need to know"
for work in the physical sciences as chemistry and physics?
The key words here are "need to know." Our model should be the police academy where my son graduated after earning a degree from San Jose State University. Police officers "need to know" a huge chunk of law to be effective in their work. As the candidate progresses through the police academy, they internalize statute after statute on a "need to know" basis. I see a similar strategy in chemistry or physics. As we move through the course, when we "need to know," the mind opens up a window. We seem to understand information in almost one exposure.
But, don't we want our children to be math-literate?
Of course we do. But how are we going to do this? Obviously, our current attempt at "forcing" the information into young learners is not working. Evidence: We spend more on remedial mathematics in America than all other math programs put together.
Now consider this: We have successful electives that attract thousands of students. The names of these courses: Art Appreciation and Music Appreciation. It is time for a new elective called Mathematics Appreciation.
In my new book, The Super School of the 21st Century, I suggest that the content of this new elective should be the dramatic stories of mathematicians. For example, there is intrigue in the story of Bertrand Russell and Alfred North Whitehead who wrote a prize-winning volume to explain why 1 + 1 = 2. How can someone write an entire book on something as obvious as 1 + 1 = 2?
Then there is Rene Descartes, the 15th century French soldier and mathematician, who discovered the "Atlantis" of the mathematical world. For centuries, mathematicians believed there was no connection between geometry and algebra. Descartes felt that his colleagues were wrong. He began to search for the mysterious connection that he believed was there, but invisible.
In his diary, Descartes wrote, " One night when I was in a deep sleep, the Angel of Truth came to me and whispered the secret connection between geometry and algebra." Without this revelation, our world as we know it, would disappear. There would be no architecture, engineering or science. All of our technological, scientific, and medical marvels were discovered because of a visit from Descartes' Angel of Truth.
Carl Friedrich Gauss, recognized as the Prince of Mathematics, wrote his thoughts in a scientific diary that is now revered as "the most precious document in all mathematics." One of his famous discoveries was to see a hidden pattern in numbers that was invisible to mathematicians for hundreds of years.
We must include in our stories the Michelangelo of science and mathematics, Sir Isaac Newton. He discovered calculus, the composition of white light, and the laws of gravity. Sir Isaac believed that God must make some personal adjustments from time to time to keep planets in their orbit. Most people do not know that Newton conducted secret experiments in alchemy, a capital offense for which people were executed in 18th century England. He was fascinated with the occult, a subject he explored in a million words written in his private notebooks.
The history of mathematicians will intrigue young people. For example, Laura Nickel and Curt Noll were only 15 years-old when they heard the story of the Chinese mathematician Chen Jin-Run. This person dedicated his professional life to exploring the fundamental theorem of arithmetic that involves prime numbers.
All numbers seem to be composed of certain other numbers called primes. What fascinated Nickel and Noll was the notion that primes are a sort of DNA of all numbers. The two high school students were surprised that no pattern has yet been found to predict the highest prime ever discovered. They set out to find that number. :
Mathematics professors warned them that their project was doomed to failure, but they vowed to prove the experts wrong. After 2,000 hours of work and 44 computer tests, they found the elusive number which was confirmed by theoretical mathematicians at the University of California's Berkeley campus.
If a student is to be wildly passionate about mathematics, the student must have the opportunity to experience the romance of mathematics. Romance comes first. Later comes the skills.
James J. Asher is the recipient of the Outstanding Professor Award in a faculty of 1,500 Ph.Ds from California's historic first public institution of higher learning, San Jose State University. Both Berkeley and UCLA were branches of San Jose State when they started. His teaching specialty is applied research statistics. This article was excerpted from his books, Brainswitching: Learning on the Right Side of the Brain and The Super School of the 21st Century: Teaching on the Right Side of the Brain published by Sky Oaks Productions, Inc., P.O. Box 1102, Los Gatos, CA 95031.
Brand New! Check out Dr. Asher's fantastic new book, TheWeirdandWonderfulWorldofMathematicalMysteries. This book includes some of the most colorful people in history such as Archimedes, Pythagoras, Euclid, Fermat, Descartes, Cauchy, Goldbach, Newton, and Einstein... who often went for days without eating or sleeping trying to decipher these mysteries--then, the excitement of discovery! You will find out how they used the right side of their brain to make spectacular breakthroughs that dramatically changed our world. Also, Dr. Asher shows how he solved two of the world's most baffling mathematical mysteries! To purchase online visit the catalog.
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Encyclopedic Dictionary of Mathematics. 4 vols. Ref. QA 5 .I8313 1987 "Intended to be a compact, up-to-date source of information comprising... all significant results in all fields of science, pure and applied, from the elementary to the advanced level." First three volumes are alphabetically arranged. Volume 4 contains tables of formulae, names and subject indexes. KL also owns the 2-volume first ed., Ref. QA 5 .N5 1977.
CRC Concise Encyclopedia of Mathematics. Ref. QA 5 .W45 1999 This book is a compendium of formulas, definitions, figures, tabulations and references.
MathWorld This is an online encyclopedia. Covers algebra, applied mathematics, history and terminology, etc. Encyclopedia of Mathematics "The Encyclopedia of Mathematics wiki is an open access resource designed specifically for the mathematics community. The original articles are from the online Encyclopaedia of Mathematics, published by Kluwer Academic Publishers in 2002. With more than 8,000 entries, illuminating nearly 50,000 notions in mathematics, the Encyclopaedia of Mathematics was the most up-to-date graduate-level reference work in the field of mathematics."
The International Dictionary of Applied Mathematics. Ref. QA 5 .I5 "Defines the terms and describes the methods in the applications of mathematics to thirty-one fields of physical science and engineering" (preface). Both practical and general terms and definitions are utilized. The foreign language indices list the French, German, Russian, and Spanish equivalent of English words.
Dictionary of Logical Terms and Symbols. Ref. QA 9 .G698 Presents "compactly, concisely, and side by side a variety of alternative notational systems currently used by logicians, computer scientists, and engineers" (preface). Also includes a glossary and a bibliography.
Mathematical Handbook for Scientists and Engineers. Ref. QA 40 .K598 1968 This book is designed as a "comprehensive reference collection of mathematical definitions, theorems, and formulae for scientists, engineers and students" and also as a means of presenting surveys of mathematical methods for non-reference applications. "Each chapter is arranged so as to permit review of an entire mathematical subject." Includes glossary of symbols and notations and tables. Handbook of Mathematical Functions
Math Archives Good for teaching materials, software; Topics in Mathematics has a searchable database.
MathDL Composed of four parts: The Journal of Online Mathematics, which includes essays on teaching strategies; Digital Classroom Resources, which "provides a select collection of free online learning materials which are available through the site. These materials have been classroom tested and peer reviewed"; Convergence, which is an online history magazine; and MAA Reviews, which is a database of books and book reviews. Must be an MAA member to use.
Multi-Repository Mathematics Collection "From the Multi-Repository Mathematics Collections site you will gain access to three of the most significant mathematics collections digitally available: The Mathematics Collection from Cornell, the Mathematics Collection from Göttingen, and the University of Michigan's Historical Math Collection." Can do basic, Boolean and bibliographic searches.
QuickMath "QuickMath is an automated service for answering common math problems over the internet.
Think of it as an online calculator that solves equations and does all sorts of algebra and calculus problems - instantly and automatically!
When you submit a question to QuickMath, it is processed by Mathematica, the largest and most powerful computer algebra package available today. The answer is then sent back to you and displayed right there on your browser, usually within a couple of seconds. " |
tNumber Theory by George E. Andrews Undergraduate text uses combinatorial approach to accommodate both math majors and liberal arts students. Covers the basics of number theory, offers an outstanding introduction to partitions, plus chapters on multiplicativity-divisibility, quadratic congruences, additivity, and more Chemical History of a Candle by Michael Faraday This highly readable text by a famous inventor explores the components and weight of the atmosphere; capillary attraction; carbon content in oxygen and living bodies; and much more. Numerous illustrationsA Short Account of the History of Mathematics by W. W. Rouse Ball This standard text treats hundreds of figures and schools instrumental in the development of mathematics, from the Phoenicians to such 19th-century giants as Grassman, Galois, and Riemann.
History of Analytic Geometry by Carl B. Boyer This study presents the concepts and contributions from before the Alexandrian Age through to Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850. 1956 edition. Analytical bibliography. Index.
The Origins of Cauchy's Rigorous Calculus by Judith V. Grabiner This text examines the reinterpretation of calculus by Augustin-Louis Cauchy and his peers in the 19th century. These intellectuals created a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. 1981 edition.
Product Description:
that we know today |
Upon successful completion of the course, the student will be able to:
1. Use the properties of real numbers to simplify and evaluate expressions.
2. Solve linear equations and inequalities.
3. Use and transform formulas and functions.
4. Graph linear equations and inequalities in one and two variables .
5. Write the three forms of the equation of a line.
6. Solve systems of linear equations by graphing, substitution, and addition.
7. Apply the laws of exponents and use scientific notation.
8. Factor and perform operations with polynomials.
9. Solve quadratic equations by three methods: factoring, completing the square ,
and using the Quadratic Formula.
10. Graph quadratic equations.
11. Perform operations with rational algebraic expressions , and solve rational
equations.
12. Simplify and perform operations with radical expressions and rational
exponents.
13. Use exponential and logarithmic functions and the properties of logarithms.
14. Solve word problems using one or more of the above skills.
Course Requirements
Students are expected to attend all scheduled classes, do the homework assigned
each day for
the next class, take tests, and be active participants in the class.
Algebra for College Students, Mark Dugopolski, 5th edition, McGraw Hill, 2009
If purchased in the SMCC bookstore, the text will be packaged with the Student
Solutions
Manual and a MathZone Access card.
Scientific calculator (required)
Non-discrimination and Disability Statements:
Southern Maine Community College is an equal opportunity/affirmative action
institution and
employer. For more information, please call 207-741-5798.
If you have a disabling condition and wish to request accommodations in order to
have reasonable
access to the programs and services offered by SMCC, you must
register with the disability services
coordinator, Mark Krogman, who can be
reached at 741-5629.
(TTD 207-741-5667) Further information about services for students with
disabilities and the
accommodation process is available upon request at this
number.
Course Evaluation: Students may evaluate the course online and anonymously by
going to
"Resources for Current Students" at the SMCC homepage and selecting
"Evaluate Your Courses."
The online course evaluation is available to students
two weeks prior to the end date of the course.
Students cannot see a course
grade online until the online course evaluation is completed.
Grading Policy: Test will count 50% towards your final grade. Homework and
quizzes will count as
the remaining 50% of the final grade.
Homework Policy: Homework will be assigned each class and will be due on the
date specified on
the assignment calendar. Do not list answers only, but show
work that is required for the solution.
Use the student solution manual as a
guide on how I would like you to show your work. Homework
will be graded as a
10-point quiz grade. The grade will be based on your performance and
completeness of the assignment. Homework submitted late will be assessed a
deduction of 2 points
for each calendar day late.
Quiz/Test Makeup Policy: Quizzes may not be made up. You must be in class the
day of a quiz to
take the quiz and get credit for it. If you are absent the day
of a test, you have 1 calendar week to
make up the test. After the week has
elapsed, a grade of zero (0) will be assigned for the test.
Attendance Policy: Attendance in class is critical to your success in this
course and is mandatory.
Three consecutive absences will result in an automatic
failure (AF), and 5 cumulative absences will
result in an automatic failure (AF). Attendance is defined
as being in class for the whole class. If you
are more than 10 minutes late or need to leave early for any reason, you will be
counted as absent for
the class. There is no such thing as an excused absence. Make your doctor
appointments and
schedule your other commitments when you are not scheduled for class.
Tutoring Service and Extra Help: This is not an easy course and we will move
very rapidly. You
must keep up to be successful. I will be available to help you by appointment.
There are also math
tutors available free of charge and at your convenience in the Academic
Achievement Center located
on the second floor of the Campus Center. I strongly suggest visiting them early
and often to keep up
and make sure you comprehend the skills and concepts we will be studying.
Required Course Topics, MAT 108:
CHAPTER 1 THE REAL NUMBERS
all sections
Sets and The Real Numbers
Operations on the Set of Real Numbers
Evaluating Expressions
Properties of the Real Numbers and Using the Properties |
MATH 211
Fundamental Mathematics I
CR:
3.0
Prerequisite: not open to first-year students. A course for prospective elementary teachers covering the methods of instruction of mathematics and emphasizing a hands-on approach. topics include number systems, elementary number theory, ratio, proportion, and percent.
Course Overview
Students in the mathematical programs analyze and solve problems in a variety of environments while improving and extending their logical skills. Major programs may be elected which emphasize abstract or applied mathematics.
A student may earn either a Bachelor of Arts or a Bachelor of Science degree in mathematics. Interdepartmental majors are offered in mathematical economics and mathematics-physics. Students interested in any of these majors are encouraged to consult the department chair for advising assistance.
Note: No more than two 300-level courses may be double-counted for a mathematics major and a statistics minor. No 300-level course may be double-counted for a mathematics minor and a statistics minor.
Teacher Licensure Students seeking teacher licensure in secondary mathematics must include MATH 310 and MATH 333 in their major program. In addition, one course in statistics (MATH 106, MATH 205, or MATH 304) must be included in the major program. |
For all your math-ing needs.
"The essence of mathematics is not to make simple things complicated, but to make complicated things simple."
Mathematics T is one of the subjects offered to the science stream STPM students in Malaysian secondary schools. The Mathematics T syllabus consists of sixteen topics to be covered in Lower Six (half a year) and Upper Six (one year), at the end of which the students will be tested in two separate papers in STPM Examination.
The topics for each paper are listed below. The order in which the topics are taught varies from school to school.
Candidates for Mathematics T are required to take both Paper 1 (954/1) and Paper 2 (954/2), each consists of twelve compulsory questions of variable marks allocations totalling 100 marks. Both papers have equal weightage, each contributing 50% to the total marks. The duration for each paper is three hours. During the examination, the candidates are allowed to use scientific calculators (which cannot be programmed) and they will be given a booklet of mathematical notations, definitions and formulae.
STPM uses a Cumulative Grade Point Average (CGPA) system, where there are 11 grades ranging from A to F. The grade points for each of these grades are shown in the following table. A candidate must score a minimum of grade C in order to pass a paper. |
Run a Quick Search on "Trigonometry For Dummies" by Mary Jane Sterling to Browse Related Products:
Short Desription
A plain-English guide to the basics of trig From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English, offering lots of easy-to-grasp example problems, and adding a dash of humor and fun. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers. Mary Jane Sterling (Peoria, IL) has taught mathematics at Bradley University in Peoria for more than 20 years. She is also the author of the highly successful Algebra For Dummies (0-7645-5325-9).
If You Enjoy "Trigonometry For Dummies (Paperback)", May We Also Recommend: |
Mathematical Communication
Using visuals
This webpage includes links to effective mathematical visuals online, including proofs without words. Also included is a list of tools for visualizing math (both commercial and free), and a list of resources both online and in the literature that present pedagogical strategies, theory, research, &/or experience, both for teaching students to communicate math via visualizations, and for using visualizations to help students learn math. |
Costs
Course Cost:
$299.00
Materials Cost:
None
Total Cost:
$299
Special Notes
State Course Code
02125Students do best when they have an understanding of the conceptual underpinnings of calculus. This course stresses the dual concepts of conceptual understanding of calculus and fluency in the procedures that accompany those concepts. If students can grasp the reasons for an idea or theorem, they can usually figure out how to apply it to the problem at hand. We will study four major ideas during the year: limits, derivatives, indefinite integrals, and definite integrals. Students practice the skills of calculus while they solve real-world problems with calculus concepts |
Introduction to Abstract Algebra with Notes to the Future Teacher, An, CourseSmart eTextbook
Description
For courses in Abstract Algebra.
Designed for future mathematics teachers as well as mathematics students who are not planning careers in secondary education, this text offers a traditional course in abstract algebra along with optional notes that connect its mathematical content to school mathematics.Elementarynumber theory and rings of polynomials are treated before group theory. Prerequisites include some experience with proof. (A brief appendix reviews certain basics of logic, proof, set theory, and functions.) Students should also have access to a Computer Algebra System (CAS), or a calculator with CAS capabilities. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
Table of Contents
1. Topics in Number Theory
2. Modular Arithmetic and Systems of Numbers
3. Polynomials
4. A First Look at Group Theory
5. New Structures from Old
6. Looking Forward and Back |
This course deals with the different ways in which data has to be
encoded in order to be handled effectively. Three main topics will be
discussed: cryptographic or secret codes; error-correction codes
used, for example, in CD's and in transmitting data through space; and
data compression codes used in reducing the amount of space needed to
store data. The course uses algebraic ideas that you have encountered
in previous years such as matrices and determinants, polynomials, and
modular arithmetic.
The prerequisites for this module are basic but essential: you must be
able to handle matrices with confidence, particularly matrix
multiplication and computing small determinants; you must also know
about linear dependence and independence; you must know the
basics of modular arithmetic; you must know the basics of polynomials;
and you must know the basics of sets. Most of these prerequisites were
covered in my first year module Algebra A. The remaining prerequisites
should be
second nature to anyone who has progressed through the first and third
years. |
Math
Pre-Algebra
This course provides students with a survey of preparatory topics for high school mathematics, including foundations for high school Algebra and Geometry.
Algebra I
This course expands on basic mathematical skills while developing the student's knowledge of the properties of the real number system and algebraic concepts. Throughout this course students also develop critical logical thinking skills. Topics include solving equations and inequalities, performing various operations with polynomials, factoring polynomials, algebraic fractions, graphing equations and inequalities, solving systems of linear equations, exponents and radicals, and quadratic equations.
Geometry
This course requires students to develop logical reasoning skills in order to use the properties of geometric figures to solve problems and write proofs. Algebraic concepts are reinforced throughout the course. Students explore the various topics with reference to realistic and relevant applications. Topics include points, lines, planes and angles, reasoning and proofs, line types, congruent triangles and relevant applications, quadrilaterals, connecting proportion and similarity, right angles, analyzing circles, polygons, surface area and volume, and coordinate geometry.
Algebra II
This course expands on topics covered in the Algebra I course. This course challenges students to continue development of logical thinking skills through a study of advanced algebraic concepts. Topics include problem solving through relations and functions (linear, polynomial, rational, quadratic, exponential and logarithmic) and graphing (linear functions, conic sections, exponential and logarithmic.
Calculus
Calculus is an elective course that covers the topics of first semester college calculus. This course covers various topics in differential and integral calculus and their applications.
AP Statistics
The purpose of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes:
Students who successfully complete the course and examination may receive credit and/or advanced placement for a one-semester introductory college statistics course. This does not necessarily imply that the high school course should be one semester long.
Pre-calculus
The purpose of this course is to adequately prepare students for calculus by exposing students to the following material: Piecewise Functions, Defined Functions (Even and Odd Functions),.Polynomials, Rational Functions, Geometric Transformations of Functions, Algebra of Functions, Composition of Functions, Trigonometric Functions, Exponential Functions, Inverses of Functions, Logarithms, Parameters and Functions, Parametric Equations, Polar Coordinates, Graphing with Technology, Solving Equations, Curve Fitting and Conic Sections |
This analyzing, and evaluating data involving one and two variables. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
MEL3E - Mathematics for Work & Everyday Life (Workplace):
This
MCR3U - Mathematics: Functions (University):
This course introduces the mathematical concept of the function by extending studentsomial and rational expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
MCF3M - Mathematics: Functions and Applications (University/College):
This course introduces basic features of the function by extending students experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modeling |
Uncover each step in the solution process as you learn ways to help your students master the skills they need to answer questions like these: "What does it mean to solve a system of linear equations?" and "What do all of the procedures used to solve such systems have in common?"
Learn how to help students grasp the symbolic representations of functions, while representing families of linear functions in multiple formats. Observe and discuss videos of students to gain insight into student thinking and explore strategies to address their misconceptions. Acquire interactive software and activities to use in the classroom that demonstrate the link between symbolic forms of linear functions and graphical forms.
Work with families of quadratic functions to help your students understand what makes a function quadratic. Explore the information conveyed by polynomials, vertex and root forms, and assist students in shifting between object and process viewpoints. As a final project, you'll create a lesson plan or action plan applying the strategies learned. |
Single Variable
Brings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. This book ...Show synopsisBrings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. This book includes the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. It also features conceptual and modeling problemsYipes! This book may be a great reference for people who know calculus but for me, just learning? It's a struggle and a challenge. It's sparse on the text -- example problems frequently skip steps or lack descriptions of what happened between step "n" and "n + 1". It suffers from a lack of illustrations |
is a classic analysis written by French mathematician Edouard Goursat. This book covers such topics as integration, differential equation and multiple integral and etc. The proof are strict, and the development of proofs are much more make sense than today's delta-epsilon proofs. The theorems in the book are proved in a much more natural and intellectual way.
This is a study book devoted to the subject of differential and integral calculus. Although the study book was published for the first time in 1869, it has been very popular since then among readers who study mathematics. The study book contains very good explanations and wonderful examples of different kinds of differential and integral calculus |
MA135 - College Algebra
Course Description: MA 135 College Algebra A consideration of those topics in algebra necessary for the calculus. Topics include: Solving equations and inequalities, graphing, functions, complex numbers, the theory of equations, exponential and logarithmic functions. Prerequisite: MA 125, or a high school or transfer course equivalent to MA 125, or an ACT math score >= 23, or an SAT math score >= 510, or a COMPASS score >= 66 in the Algebra placement domain, or a COMPASS score 0-45 in the College Algebra placement domain. 3:0:3@ (From catalog 2010-2011) |
Mathematics
"We may in fact regard geometry as the most ancient branch of physics. Without it I would have been unable to formulate the theory of relativity…" Albert Einstein (1921)
The importance of Mathematics in the school curriculum had finally been recognised by the National Department of Education with the introduction of the new FET syllabus in 2006. As from 2006, Mathematics or Mathematical Literacy became a compulsory subject from grades 10-12. In 2008 the first set of matriculants emerged from this new FET curriculum. Fortunately for us at Glenwood High School this is not a foreign concept as Mathematics has always been a compulsory subject from grades 8 -12.
There are now three examination papers in the matric final examination, viz. Paper I and Paper II which are compulsory and Paper III which is optional. At Glenwood High School Mathematics Paper III is compulsory for all learners' in grades 10 and 11 who take Mathematics. In grade 12, the boys' are allowed the choice of writing Paper III in the final examination.
Our boys also participate in the Harmony Gold South African Mathematical Olympiad. 100 of our top pupils are entered at no cost.
We have a Mathematics Club which meets on one afternoon a week. Here the boys are exposed to higher order Mathematics problems which will help them in the Mathematics Olympiad as well as the AMESA Mathematics Challenge.
There are extra lessons offered in the afternoon for all grades from grade 8 to grade 12 as well as for Paper III (Grade 12). The boys' are given every opportunity to improve and excel in Mathematics. We hope that they all take up the challenge! |
an expanded version of Calculus and its Applications, Tenth Edition, by Bittinger, Ellenbogen, and Surgent. This edition adds coverage of trigonometric functions, differential equations, sequences and series, probability distributions, and matrices.
Calculus and Its Applications has become a best-selling text because of its accessible presentation that anticipates your needs. The writing style provides intuitive explanations that build on e... MOREarlier mathematical experiences. Explanations are often coupled with figures to help you visualize new calculus concepts. Additionally, the text's numerous and up-to-date applications from business, economics, life sciences, and social sciences help motivate you. Algebra diagnostic and review material is available for those who need to strengthen basic skills. Every aspect of this text is designed to motivate and help you to more readily understand and apply the mathematics.
2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
2.2 Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs
2.3 Graph Sketching: Asymptotes and Rational Functions
2.4 Using Derivatives to Find Absolute Maximum and Minimum Values
2.5 Maximum-Minimum Problems; Business and Economic Applications
2.6 Marginals and Differentials
2.7 Implicit Differentiation and Related Rates
3. Exponential and Logarithmic Functions
3.1 Exponential Functions
3.2 Logarithmic Functions
3.3 Applications: Uninhibited and Limited Growth Models
3.4 Applications: Decay
3.5 The Derivatives of ax and loga x
3.6 An Economics Application: Elasticity of Demand
4. Integration
4.1 Antidifferentiation
4.2 Antiderivatives as Areas
4.3 Area and Definite Integrals
4.4 Properties of Definite Integrals
4.5 Integration Techniques: Substitution
4.6 Integration Techniques: Integration by Parts
4.7 Integration Techniques: Tables
5. Applications of Integration
5.1 An Economics Application: Consumer Surplus and Producer Surplus
5.2 Applications of Integrating Growth and Decay Models
5.3 Improper Integrals
5.4 Numerical Integration
5.5 Volume
6. Functions of Several Variables
6.1 Functions of Several Variables
6.2 Partial Derivatives
6.3 Maximum-Minimum Problems
6.4: An Application: The Least-Squares Technique
6.5 Constrained Optimization
6.6 Double Integrals
7. Trigonometric Functions
7.1 Basics of Trigonometry
7.2 Derivatives of Trigonometric Functions
7.3 Integration of Trigonometric Functions
7.4 Inverse Trigonometric Functions and Applications
8. Differential Equations
8.1 Differential Equations
8.2 Separable Differential Equations
8.3 Applications: Inhibited Growth Models
8.4 First-Order Linear Differential Equations
8.5 Higher-Order Differential Equations and a Trigonometry Connection
9. Sequences and Series
9.1 Arithmetic Sequences and Series
9.2 Geometric Sequences and Series
9.3 Simple and Compound Interest
9.4 Annuities and Amortization
9.5 Power Series and Linearization
9.6 Taylor Series and a Trigonometry Connection
10. Probability Distributions
10.1 A Review of Sets
10.2 Probability
10.3 Discrete Probability Distributions
10.4 Continuous Probability Distributions
10.5 Mean, Variance, Standard Deviation, and the Normal Distribution
11. Systems and Matrices (online)
11.1 Systems of Linear Equations
11.2 Gauss-Jordan Elimination
11.3 Matrices and Row Operations
11.4 Matrix Arithmetic: Equality, Addition and Scalar Multiples
11.5 Matrix Multiplication, Multiplicative Identities and Inverses
11.6 Determinants and Cramer's Rule
11.7 Systems of Linear Inequalities and Linear Programming
Marvin Bittinger has been teaching math at the university level for more than thirty-eight years. Since 1968, he has been employed at Indiana University Purdue University Indianapolis, and is now professor emeritus of mathematics education. Professor Bittinger has authored over 190 publications on topics ranging from basic mathematics to algebra and trigonometry to applied calculus. He received his BA in mathematics from Manchester College and his PhD in mathematics education from Purdue University. Special honors include Distinguished Visiting Professor at the United States Air Force Academy and his election to the Manchester College Board of Trustees from 1992 to 1999. His hobbies include hiking in Utah, baseball, golf, and bowling. Professor Bittinger has also had the privilege of speaking at many mathematics conventions, most recently giving a lecture entitled "Baseball and Mathematics." In addition, he also has an interest in philosophy and theology, in particular, apologetics. Professor Bittinger currently lives in Carmel, Indiana, with his wife, Elaine. He has two grown and married sons, Lowell and Chris, and four granddaughters.
David Ellenbogen has taught math at the college level for twenty-two years, spending most of that time in the Massachusetts and Vermont community college systems, where he has served on both curriculum and developmental math committees. He has also taught at St. Michael's College and the University of Vermont. Professor Ellenbogen has been active in the Mathematical Association of Two Year Colleges since 1985, having served on its Developmental Mathematics Committee and as a delegate, and has been a member of the Mathematical Association of America since 1979. He has authored dozens of publications on topics ranging from prealgebra to calculus and has delivered lectures at numerous conferences on the use of language in mathematics. Professor Ellenbogen received his BA in mathematics from Bates College and his MA in community college mathematics education from the University of Massachusetts at Amherst. A cofounder of the Colchester Vermont Recycling Program, Professor Ellenbogen has a deep love for the environment and the outdoors, especially in his home state of Vermont. In his spare time, he enjoys playing keyboard in the band Soularium, volunteering as a community mentor, hiking, biking, and skiing. He has two sons, Monroe and Zack.
Scott Surgent received his B.S. and M.S. degrees in mathematics from the University of California—Riverside, and has taught mathematics at Arizona State University in Tempe, Arizona, since 1994. He is an avid sports fan and has authored books on hockey, baseball, and hiking. Scott enjoys hiking and climbing the mountains of the western United States. He was active in search and rescue, including six years as an Emergency Medical Technician with the Central Arizona Mountain Rescue Association (Maricopa County Sheriff's Office) from 1998 until 2004. Scott and his wife, Beth, live in Scottsdale, Arizona. |
Functions and Their Graphs. Using Technology: Graphing a Function. The Algebra of Functions. Portfolio. Functions and Mathematical Models. Using Technology: Finding the Points of Intersection of Two Graphs and Modeling. Limits. Using Technology: Finding the Limit of a Function. One-Sided Limits and Continuity. Using Technology: Finding the Points of Discontinuity of a Function. The Derivative. Using Technology: Graphing a Function and Its Tangent Line. Summary of Principal Formulas and Terms. Review Exercises.
3. DIFFERENTIATION.
Basic Rules of Differentiation. Using Technology: Finding the Rate of Change of a Function. The Product and Quotient Rules. Using Technology: The Product and Quotient Rules. The Chain Rule. Using Technology: Finding the Derivative of a Composite Function. Marginal Functions in Economics. Higher-Order Derivatives. Using Technology: Finding the Second Derivative of a Function at a Given Point. Implicit Differentiation and Related Rates. Differentials. Portfolio. Using Technology: Finding the Differential of a Function. Summary of Principal Formulas and Terms. Review Exercises.
4. APPLICATIONS OF THE DERIVATIVE.
Applications of the First Derivative. Using Technology: Using the First Derivative to Analyze a Function. Applications of the Second Derivative. Using Technology: Finding the Inflection Points of a Function. Curve Sketching. Using Technology: Analyzing the Properties of a Function. Optimization I. Using Technology: Finding the Absolute Extrema of a Function. Optimization II. Summary of Principal Terms. Review Exercises.
Antiderivatives and the Rules of Integration. Integration by Substitution. Area and the Definite Integral. The Fundamental Theorem of Calculus. Using Technology: Evaluating Definite Integrals. Evaluating Definite Integrals. Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions. Area between Two Curves. Using Technology: Finding the Area between Two Curves. Applications of the Definite Integral to Business and Economics. Using Technology: Consumers' Surplus and Producers' Surplus. Summary of Principal Formulas and Terms. Review Exercises.
Functions of Several Variables. Partial Derivatives. Using Technology: Finding Partial Derivatives at a Given Point. Maxima and Minima of Functions of Several Variables. The Method of Least Squares. Using Technology: Finding an Equation of a Least-Squares Line. Constrained Maxima and Minima and the Method of Lagrange Multipliers. Double Integrals. Summary of Principal Terms. Review Exercises. Answers to Odd-Numbered Exercises.
0534419860 Purchased as new and in great condition. We cannot guarantee the availability of CD/DVD or other resource materials such as access code etc if the book is so described by the publisher unle...show moress we indicate their availability in our own description. Will ship now if ordered before 2pm cst ...show less
05344198301.19 |
97803958940Mathematics for Elementary School Teachers
Intended for the one- or two-semester course required of Education majors, MATHEMATICS FOR ELEMENTARY SCHOOL TEACHERS, 5E, offers future teachers a comprehensive mathematics course designed to foster concept development through examples, investigations, and explorations. Visual icons throughout the main text allow instructors to easily connect content to the hands-on activities in the corresponding Explorations Manual. Bassarear presents real-world problems, problems that require active learning in a method similar to how archaeologists explore an archaeological find: they carefully uncover the site, slowly revealing more and more of the structure. The author demonstrates that there are many paths to solving a problem, and that sometimes, problems have more than one solution. With this exposure, future teachers will be better able to assess student needs using diverse approaches |
Lulu Marketplace
Boolean Algebra at School, Vol 1
The main focus of this book is on actively engaging students in the mathematical processes of modeling and axiomatization. The book starts off with some challenging problems involving switching circuits. Students are then engaged in gradually developing a mathematical model in order to solve these problems. Suitable notation is firstly introduced for series and parallel switches, and then followed by truth tables, and the discovery of several interesting mathematical properties related to switching circuits.
After development of the model and solution of the original problems, a section on systematization follows where students are led through some proof activities to identify a suitable set of axioms. The book concludes with a short historical overview of the development and application of Boolean Algebra. |
Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12
32.
Web Site S.O.S. Mathematics - Calculus Check out a good list of calculus problems with solutions. This is a free resource for math review material from Algebra to Differential Equations!
Web Site Order of Operations When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed. The phrase PEMDA (Parenth... Curriculum: Mathematics Grades: 5, 6, 7, 8
37.
Web Site Intermediate Algebra Tutorials 42 Tutorials that math teachers can use with student or students can work on their own to reinforce skills, as homework, or review during class. Tutorials in... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12, Junior/Community College, University
38.
Web Site Variables This site covers symbol variables and substitution of symbols to discover unknown values. In simple terms it shows you how a box is waiting for a value. (Key... Curriculum: Mathematics Grades: 6, 7, 8
39.
Web Site Introduction to Algebra Think Algebra is hard? Think again - this site explains the history along with simple equations. Each paragraph scaffolds skills until you get it. Than at th... Curriculum: Mathematics Grades: 3, 4, 5, 6 |
If you can teach game theory, that could be good. It's bread and butter for mathematical economics and political science (even ecologists learn it now) -- I think the subject illustrates the point that math is not limited in application to situations which involve numbers. In addition to being useful, it's very elementary to solve games (although the fundamental fact that mixed strategy Nash equilibria exist requires topology to prove, it doesn't provide an algorithm for finding them -- actually solving games is more combinatorial). Proving that sets of strategies are/are not Nash equilibria can introduce students to the concept of a formal mathematical proof in a setting which I think is straightforward.
Unfortunately, I can't think of a textbook that would be good, but maybe someone else knows one. |
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This book provides a pedagogical and comprehensive introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings...
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A theft and a hold-up, an impostor trying to collect an inheritance, the disappearance of a lab mouse worth several hundred thousand dollars, and a number of other cases : these are the investigations led by...
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It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers... |
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Beginning Math and Physics for Game Programmers
Description
Coursetextbook, students can search the text, make notes online, print out reading assignments that incorporate lecture notes, and bookmark important passages for later review. For more information, or to subscribe to the CourseSmart eTextbook, visit
Back Cover
Whether you're a hobbyist or a budding game design pro, your objective is probably the same: To create the coolest games possible using today's increasingly sophisticated technology. To do that, however, you need to understand some basic math and physics concepts. Not to worry: You don't need to go to night school if you get this handy guide! Through clear, step-by-step instructions, author Wendy Stahler covers the trigonometry snippets, vector operations, and 1D/2D/3D motion you need to improve your level of game development. Each chapter includes exercises to make the learning stick, and Visualization Experience sections are sprinkled throughout that walk you through a demo of the chapter's content. By the end of the volume, you'll have a thorough understanding of all of the math and physics concepts, principles, and formulas you need to control and enhance your user's gaming experience.
Author
Wendy Stahler was the first course director of the Game Design and Development Program at Full Sail Real World Education in Orlando, Florida. During her six years at Full Sail, she concentrated much of her time toward developing the math and physics curriculum. Wendy is also an adjunct professor at Rollins College in the IT department, and just recently took on her next challenge of IT training in the corporate world. Wendy graduated from Rollins College earning an Honors B.A. in Mathematics with a concentration in Computer Science and an MA in Corporate Communication and Technology, graduating with honors. |
Tensor analysis is an essential tool in any science (e.g. engineering,
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MATH 3795 Introduction to Comp. Math.October 20, 2008Assignment 51. (10 points) Use fzero to try to find a zero of each of the following functions in the given interval. Do you see any interesting or unusual behavior? (a) atan(x) - /3 on [0, 5
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as these are covered very incompletely these days. |
Navigation Tree
Teaching Material
See also the page on
Learning GAP,
which refers to material that may help you if you want to learn
GAP on your own or to teach it.
GAP has been used in lecture courses of various levels,
both for providing examples and for creating teaching material, in
particular exercises. However, at present we know of rather few examples
of such teaching materiæl being made available to the public. We therefore
ask to inform us if you have such material and are willing to share it
with others. The person to contact is
Edmund Robertson at St Andrews who
has agreed to maintain and extend this page.
A lab manual
Abstract Algebra with GAP
by Julianne Rainbolt and
Joseph A. Gallian
containing a collection of exercises that use GAP
and are appropriate for a first course in abstract algebra.
This manual was originally developed to be used with Gallian's book
Contemporary Abstract Algebra).
Lectures and Workshops on Groups, Applications, and
GAP by
Alice Niemeyer,
held in September 2004 at the University of Malaya at Kuala Lumpur.
The course describes applications of GAP
to counting and randomised algorithms.
Graph Isomorphism Problem
by Vincent Remie, Eindhoven University of Technology.
This looks at ways to show that two graphs on n vertices
are not isomorphic. Code is given for a number of GAP
functions to examine graphs.
Information on usage of CommSemi and other semigroups functionality
in GAP can be found in "Tutorial - Computing with
semigroups in GAP" by
Isabel M. Araújo and
Andrew Solomon,
which is available
here.
Teaching material in Japanese:
The home page of
Toshiaki Shoji,
in its section 'Refresh Corner' provides links to PDF as well as
PostScript versions of two Japanese texts 'How to play GAP'
(dating from Oct. 2002 and Feb 2005, resp.) that contain parts
of the GAP tutorial with additional examples.
The package
ITC can be used to demonstrate the
working of some coset table methods. Examples are given in the
ITC manual. |
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Authors
Abstract
Problem-solving processes should extend beyond mere working problems by type where students are provided
algorithmic approaches to fit situations (e.g., rate, mixture, coin, investment, work) since reducing these typical problem
situations to "algorithmic processing" is counter-productive relative to higher-level problem-solving goals (NCTM, 1989,1991,
1996). By incorporating technological tools (CABRI Geometry II, spreadsheets, and graphing calculators) coupled with the
problem solving principles espoused by George Polya (famous mathematician and teacher of mathematics and mathematics
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Synopsis
Translated into many languages, this book has been the standard university-level text for decades. Revised and enlarged by the author in 1952, it offers today's students exercises in construction problems, similitude, and homothecy, properties of the triangle and the quadrilateral, harmonic division, and circle and triangle geometry |
Pages
Wednesday, May 1, 2013
This
column continues my report on results of the MAA National Study of
Calculus I, Characteristics
of Successful Programs in College Calculus.
This month I am
sharing what we learned about the use of graphing calculators (with
or without computer algebra systems) and computer software such as
Maple
or Mathematica.
Our results draw on three of the surveys:
Student
survey at start of term: We asked students how calculators and/or
computer algebra systems (CAS) were used in their last high school
mathematics class and how comfortable they are in using these
technologies.
Student
survey at end of term: We asked students how calculators or CAS had
been used both in class and for out of class assignments.
Instructor
survey at start of term: We asked instructors what technologies
would be allowed on examinations and which would be required on
examinations.
Our
first question asked students how calculators were used on exams in
their last high school mathematics class (see Figure 1). As in
previous columns, "research" refers to the responses of students
taking Calculus I at research universities (highest degree in
mathematics is doctorate), "undergrad" refers to undergraduate
colleges (highest degree is bachelor's), "masters" to masters
universities (highest degree is masters), and "two-year" to
two-year colleges (highest degree is associate's).
There
are several interesting observations to be made from this graph.
First, not surprisingly, almost all Calculus I students reported
having used graphing calculators on their exams at least some of the
time ("always" and "sometimes" were mutually exclusive
options). Second, there is a difference by type of institution.
Students at undergraduate colleges were most likely to have used
graphing calculators on high school exams (94%), then those at
research universities (91%), then masters universities (86%), and
finally two-year colleges (77%). The differences are small but
statistically significant. My best guess is that these are
reflections of the economic background of these students. A second
observation is that for most students, access to a graphing
calculator was not always allowed. However, it is still common
practice in high schools (roughly one-third of all students) to
always allow students to use graphing calculators on mathematics
exams.
Another
striking observation from Figure 1 is that the percentage of students
who were always allowed to use graphing calculators on exams is
almost identical to the percentage of students who were always
allowed to use graphing calculators with CAS capabilities on exams.
For all categories of students, over half of them were allowed to use
graphing calculators with CAS capabilities at least some of the time,
which suggests that over half of the students in college Calculus I
own or have had access to such calculators.
The
next graph (Figure 2) shows how students at the start of the term
reported their comfort level with using graphing calculators or
computer algebra systems (Maple
and Mathematica
were provided as examples of what we meant). The most interesting
feature of this graph is that students at two-year colleges are much
more likely to be comfortable with Maple
or Mathematica
than those at four-year programs. I suspect that the reason behind
this is that most Calculus I students at two-year colleges are
sophomores who took pre-calculus at that college the year before.
This gave them more opportunity to experience these computer algebra
systems.
Figure
2. Student attitude toward use of graphing calculator or CAS on a
computer such as Maple
or Mathematica.
The
graphs in Figures 3–5 show what students reported at the end of the
term about use of technology. For the graph in Figure 3, students
were asked how frequently each of these occurred in class. Percentage
shows the fraction of students who responded "about half the class
sessions," "most class sessions," or "every class session."
We note large differences in instructor use of technology generally
(for this question, "technology" was not defined), and especially
sharp differences for instructor use of graphing calculators or CAS
(with Maple and
Mathematica given as
examples). It is interesting that students are most likely to
encounter computer algebra systems in undergraduate and two-year
colleges, much less likely in masters and research universities.
Figure
3. End of term student reports on frequency of use of technology (at
least once/month). For this question, CAS refers to a computer
algebra system on a computer, such as Maple
or Mathematica.
The
first two sets of bars in Figure 4 show student responses to "Does
your calculator find the symbolic derivative of a function?" The
first set gives the percentage responding "N/A, I do not use a
calculator." The second set displays the percentage responding
"yes." Looking at the complement of these two responses, we see
that across all types of institutions, roughly 50% of students taking
Calculus I own a graphing calculator without CAS capabilities. The
third set records the percentage responding "yes" to the
question, "Were you allowed to use a graphing calculator during
your exams?" Note that there are some discrepancies between what
students and instructors report about allowing graphing calculators
on exams (Figures 4 and 6), but the basic pattern that graphing
calculators are allowed far less frequently at research universities
than at other types of institutions is consistently demonstrated.
Figure
4. End of term student reports on calculator use. No calculator = do
not use a calculator. Calculator with CAS = use a calculator with CAS
capabilities. Calc allowed on exams = graphing calculators were
allowed on exams.
We
also asked how often "The assignments completed outside of class
time required that I use technology to understand ideas." Again, we
see much less use of technology at research universities, the
greatest use at undergraduate and two-year colleges.
Figure
5. Frequency with which technology (either graphing calculators or
computers) was used for out of class assignments. Almost never = less
than once per month (includes never). Sometimes = at least once per
month but less than once per week. Often = at least once per week.
The
last two graphs (Figures 6 and 7) are taken from the instructor
responses at the start of the term: what technology they would allow
on their exams and what technology they would require on their exams.
Again, we see a clear indication that technology, especially the use
of graphing calculators without CAS capabilities, is much less common
at research universities than other types of institutions.
It
is interesting to observe that there are large numbers of instructors
who allow but do not require technology on the exams. At research
universities, 26% require the use of some kind of technology, and a
further 25% allow but do not require the use of some sort of
technology. For undergraduate colleges, 38% of instructors require
technology, an additional 42% allow it. At masters universities, 42%
require, and a further 33% allow. At two-year colleges, 52% require,
and an additional 36% allow.
Figure
6. Start of term report by instructor of intended use of technology
on exams. GC = graphing calculator. Most of those who checked "other"
reported that they allowed graphing calculators on some but not all
parts of the exam. Some reported allowing only scientific
calculators.
Figure
7. Start of term report by instructor of intended use of technology
on exams. GC = graphing calculator. Most of those who checked "other"
reported that they required graphing calculators on some but not all
parts of the exam. Some reported requiring only scientific
calculators.
We
see a pattern of very heavy use of graphing calculators in high
schools, driven, no doubt, by the fact that students are expected to
use them for certain sections of the Advanced Placement Calculus
exams. They are still the dominant technology at colleges and
universities, but there the use is as likely to be voluntary as
required. This implies that in many colleges and universities
questions and assignments are posed in such a way that graphing
calculators confer little or no advantage. The use of graphing
calculators at the post-secondary level varies tremendously by type
of institution. Yet even at the research universities, over half the
instructors allow the use of graphing calculators for at least some
portions of their exams.
Monday, April 1, 2013
Last month (MAA Calculus Study: Good Teaching) I discussed the student-described attributes
of instructors that were highly correlated with improvements in student
confidence, enjoyment of mathematics, and desire to continue to study
mathematics. This month I will discuss a second set of instructor attributes
that we are labeling "Progressive Teaching" because they are generally
associated with approaches to teaching and learning that focus on active
engagement of the students.
Here the evidence for improved results is less clear. In
particular, Sadler and Sonnert discovered a strong interaction with the
attributes we are calling "Good Teaching": teachers who rated high on Good
Teaching improved student outcomes if they also rated high on Progressive
Teaching. But if they rated low on Good Teaching, then a high rating on
Progressive Teaching had a strongly negative effect on student confidence. This
might have been expected. Good Teaching
describes student-teacher interactions, including the degree to which students
feel encouraged to participate in class and supported by the instructor. It is not surprising that students who are
encountering unfamiliar approaches to classroom learning react negatively if
they believe that that the instructor is not encouraging or supportive.
We also have evidence of some consistently positive effects
from Progressive Teaching. Even with a low score on Good Teaching, Progressive
Teaching was seen to be helpful in convincing students to continue the study of
mathematics. Our conclusions are that:
b.Good Teaching is more important to student persistence
than Progressive Teaching,
c.both can serve to improve student outcomes, and
d.teaching is most effective when instructors rate
high on both measures.
There were 12 student responses that clustered into what
we are calling Progressive Teaching:
My calculus instructor frequently 1.Assigned sections of the textbook to read before
coming to class.
2.Had students work with one another. 3.Had students give presentations. 4.Asked students to explain their thinking in
class. 5.Required students to explain their thinking on
homework assignments. 6.Required students to explain their thinking on
exams. 7.Held whole class discussions.
My calculus instructor did not frequently 8.Lecture.
Assignments completed outside of class 9.Required that I solve word problems. 10.Were
problems unlike those done in class or in the book. 11.Were
often submitted as a group project. 12.Were
returned with helpful feedback and comments.
With one exception, the following graphs show the percentage
of students who reported that their instructors employed each of these
practices often or very often (a 5 or 6 on a Likert scale from 1 = not at all
to 6 = very often). The exception is practice #8. Here we record the percentage
of students who responded 1, 2, or 3 on the same scale to the question, "During
class time, how frequently did your instructor lecture?".
We see that for most of the instructor behaviors (practices
1 through 8), the undergraduate colleges and two-year colleges are where these
are most likely to be employed. The relatively large percentage of instructors
at masters universities who had students give presentations in class (13% as
opposed to 6% at all other types of institutions) is still small and may be an
artifact of the relatively small number of responses from students at masters
universities (305 students at 18 institutions).
The research universities are where we find the most challenging
problems being posed on assignments, either word problems or those unlike those
done in class or in the book. Instructors at two-year colleges provide the most
helpful feedback on assignments, instructors at research universities the least
helpful feedback.
Friday, March 1, 2013
One of the primary goals of the MAA Calculus Study, Characteristics of Successful Programs in College Calculus (NSF #0910240), has been to
identify the factors that are highly correlated with an improvement in student
attitudes from the start to the end of the calculus course: confidence in
mathematical ability, enjoyment of mathematics, and desire to continue the
study of mathematics. To this end, Phil Sadler and Gerhard Sonnert of the
Science Education Department within the Harvard-Smithsonian Center for
Astrophysics constructed a hierarchical linear model from our survey responses
to identify these factors. The factors reside at three levels: institutional,
classroom, and individual student. Not surprisingly, most of the variation in
student attitudes can be explained by student background, but there are
influences at the institutional and classroom level. We have been particularly
interested in what happens at the classroom level where there is the greatest
opportunity for improvement.
Sadler and Sonnert ran a factor analysis of the
classroom-level variables, clumping those responses that were highly correlated.
They discovered that the responses broke into three distinct clusters, which we
are labeling "technology," "progressive teaching," and "good teaching" because
these seem to describe the characteristics of the instruction. By far, the most
important of these in terms of high correlation with improved attitudes is
"good teaching." Listed below are the 21 student-reported characteristics of
instruction that are highly correlated with each other and highly correlated
with improvements in student attitudes, characteristics that collectively we
are calling "good teaching":
My calculus instructor:
Asked questions to determine if I understood what was being
discussed.
Listened carefully to my questions and comments.
Discussed applications of calculus.
Allowed time for me to understand difficult ideas.
Helped me become a better problem solver.
Encouraged students to enroll in Calculus II.
Acted as if I was capable of understanding the key ideas of
calculus.
Made me feel comfortable asking questions during class.
Encouraged students to seek help during office hours.
Presented more than one method for solving problems.
Made class interesting.
Provided explanations that were understandable.
Was available to make appointments outside of
office hours, if needed.
My calculus instructor did not:
Discourage me from wanting to continue taking calculus.
Make students feel nervous during class.
My instructor often or very often:
Showed how to work specific problems.
Asked questions.
Prepared extra material to help students understand calculus
concepts or procedures.
In addition:
My calculus exams were a good assessment of what I learned.
My exams were fairly graded.
My homework was fairly graded.
The good news is that most calculus instructors rated highly
on most of these characteristics. This good news needs to be tempered by two
facts: Instructors could and in many cases did elect not to participate even
though other instructors at their institution were involved in the study, and these
responses were all collected at the end of the term. They reflect the opinions
of the students who had successfully navigated this course, predominantly
students who were earning an A or a B in the course (roughly 40% A, 40% B, 20%
C).
It is interesting and informative to see how students at
different types of institutions rated their instructors on these criteria. We
followed CBMS in categorizing post-secondary institutions by the highest
mathematics degree offered at that institution. I am using "research" to
designate universities that offer a PhD in Mathematics (predominantly large
state flagship universities), "masters" if the highest degree is a master's
(predominantly public comprehensive universities), "undergrad" if it is a
bachelor's degree (predominantly private liberal arts colleges), and "two-year"
if it is an associate's degree (predominantly community and technical colleges).
As shown in the graphs at the end of this article, instructors at research
universities got the lowest ratings on every characteristic except "showed how
to work specific problems." For most of these characteristics, instructors at undergraduate
colleges were the next lowest, then masters universities, and most of the time
instructors at two-year colleges received the highest ratings.
There were a few notable exceptions. Instructors at undergraduate
colleges received the highest ratings in some of the areas where one would
expect them to be strong:
Acted as if I was capable of understanding the key ideas of
calculus.
Encouraged students to seek help during office hours.
Was available to make appointments outside of office hours,
if needed.
Did not make students feel nervous during class.
Masters universities scored highest in often or very often
showing how to work specific problems, and just barely edged out two-year
colleges in "listened carefully" and "my exams were fairly graded."
There are a number of possible explanations for the
weaknesses of research universities and the strengths of two-year colleges. One
is class size. The largest classes are found at the research universities where
average class size is 53, the smallest at two-year colleges where the average
is 21. However, average class size at masters universities is larger than at
undergraduate colleges, so class size cannot be the only explanatory variable. Some
of the discrepancies between institution types may be explained by student
expectations. This is because SAT scores and high school mathematics GPA are
highest for research universities, then undergraduate colleges, then masters
universities, and lowest for two-year colleges. Better students may have higher
expectations of their instructors, or they may be more discouraged by
encountering difficulties in this course. The differences may also have
something to do with age and thus maturity of the students. The youngest
students are at research universities, the oldest at two-year colleges. They
also may be related to the relatively large number of instructors at research
universities who teach calculus but have little or no interest in teaching this
course, as opposed to two-year colleges where the interest is very high (see my
November column, MAA Calculus Study: The Instructors). Nevertheless, it is discouraging that
students at research universities seem to be getting calculus instruction that has
a worse effect on student attitudes than instruction at other types of
institutions.
Friday, February 1, 2013
The National Research Council of the National Academies has
just released the preliminary version of its report, The Mathematical Sciences in 2025[1]. This was produced in response to a request from the National
Science Foundation. It comes as the latest in a series of glimpses into the
future of mathematics that go back to the "David reports" of 1984 and 1990 [2,3]
and the "Odom study" of 1998 [4]. This report is important because it will
influence the direction NSF takes as it plans for the future.
The emphasis of the report is on the central role that the
mathematical sciences are taking within research in areas as diverse as
biology, finance, and climate science. Traditional disciplinary boundaries are
blurring. There is an increasing need for scientists who are well grounded in
mathematical sciences, especially the statistical and computational sciences,
as well as other disciplines. This goes two ways. It means opening courses and
programs in the mathematical sciences, especially at the graduate level, to
those in other fields of study, and it means ensuring that students graduating in
the mathematical sciences are prepared to work in this interdisciplinary world.
This has implications right down the line of mathematics
education. The authors of the report question whether, in a scientific world
that is dominated by big data and the challenges of large-scale computation,
the traditional calculus-focused curriculum is the most appropriate for all
students. As they say, "Different pathways are needed for students who may go
on to work in bioinformatics, ecology, medicine, computing, and so on. It is
not enough to rearrange existing courses to create alternative curricula; a redesigned offering of courses and majors
is needed [my emphasis]." (NRC 2013, p. S-9)
The report also stresses the importance of attracting more
women and students from traditionally underrepresented minorities to the mathematical
societies. This is the one place where I disagree with the report, for it
asserts that, "While there has been progress in the last 10–20 years, the
fraction of women and minorities in the mathematical sciences drops with each
step up the career ladder." (NRC 2013, p. S-10). I don't question the drop. I
question whether there has been progress over the last 10–20 years.
If we look at mathematics majors (bachelor's degrees) by
gender, we see that over the period 1990 to 2011 the number of men majoring in
mathematics grew by 25% while the number of women grew by only 10% (Figure 1).
As a result, the percentage of bachelor's degrees in the mathematical sciences
going to women has dropped to 43.1%, the first time it has been this low since
1981. This is having knock-on effects for graduate programs. The percentage of
bachelor's degrees in mathematics that went to women peaked in 1999 at 47.8%.
The percentage of master's degrees in mathematics that went to women peaked in
2004 at 45.1% and has since dropped back to 40.9%. The percentage of doctoral
degrees in mathematics that went to women peaked in 2008 and '09 at 31.0%. It
has since dropped back to 28.6%. The good news is that the past decade has seen
strong growth in the number of mathematics majors, but two-thirds of the growth
since 2001 has been in the number of men.
We see an even more discouraging pattern among Black students
(Figure 2). The number of Black mathematics majors is essentially back to where
it was twenty years ago despite the number of bachelor's degrees earned by
Black students almost tripling over this period. The number of Black
mathematics majors peaked in 1997 at 1,089. It was back down to only 840 in
2011. The number of ethnically Asian mathematics majors has been growing
strongly over the past decade. Even so, the number earning undergraduate degrees
in the mathematical sciences has only doubled since 1990, while the number
earning bachelor's degrees has tripled. The growth in the number of Hispanic
mathematics majors looks good, having slightly more than tripled in twenty
years, until you realize that the number of Hispanic students graduating from
college is almost five times what it was in 1990 (154,000 versus 33,000). Where
we do see strong growth, especially since 2007, is in the number of
non-resident aliens majoring in mathematics, which now stands at 7% of all US
mathematics majors.
I must emphasize that the NRC report does highlight the
importance of increasing the participation of women and members of
underrepresented groups. It includes the following specific recommendation:
Recommendation
5-4:
Every academic department in the mathematical sciences should explicitly
incorporate recruitment and retention of women and underrepresented groups
into the responsibilities of the faculty members in charge of
the undergraduate program, graduate program, and faculty hiring and
promotion. Resources need to be provided to enable departments to adopt,
monitor and adapt successful recruiting and mentoring programs that have
been pioneered at other schools and to find and correct any disincentives
that may exist in the department. (NRC 2013, p. 5-18)
I have only touched on a few of the topics covered in the NRC
report. It also discusses the increasingly important role of the mathematical
sciences institutes, the issue of maintaining online repositories of mathematical
research such as arXive, and the threats to
mathematics departments as more instruction—especially for the service courses
that often provide the justification for a large mathematics faculty—is moved
online. This is a report well worth reading and pondering.
Tuesday, January 1, 2013
Two things happened in the week before Christmas that got me
thinking about grade inflation. The first was that I graded the final exams for
my multivariable calculus class. I have never before seen my students do so
well. Out of 33 students in the class, 22 received an A. For my class, an A
requires earning more than 92% of the total possible grade. The last time I
graded on a curve was over 20 years ago.
This past semester I had worked these students hard. They
were responsible for and graded on:
Reading Reflections (three times per week, reading the
section and answering questions about the material before we discussed it in
class).
Two sets of homework each week (about 12 fairly
straightforward questions on WeBWorK due on Thursdays and three challenging
multi-part problems due on Mondays).
Seven short projects developed by Tevian Dray and Corinne
Manogue as part of their Bridge Project (see
These were started in groups of three or four, but each student was responsible
for writing his or her own three to five page report of the solution. For the
first report, I required a first draft that was critiqued and returned for
revision and resubmission.
A major project based on the Hydro-Turbine Optimization
chapter in Applications of Calculus
[1]. The project was started in groups. Each student was responsible for an
8–12 page paper explaining the solution. The papers were turned in, critiqued,
and returned for revision and resubmission. LaTeX and pdf files of my version
of this project are available here.
Two examinations during the semester and a final exam. After
each exam during the semester, students were required to write about the
problems they had missed points on, explain what they did wrong, and explain
how to do it correctly. They could earn back half the points they had lost. For
the final exam, they had to explain what they were doing to solve the problems,
not just give an answer.
I was available to my students every afternoon, and I also had
a great undergraduate preceptor (teaching assistant) who held help sessions
Sunday and Thursday evenings, before the homework assignments were due. By the
end of the semester, over half the class was coming to each of these, and so
she organized them into groups working with each other on the homework while
she circulated to help the groups that were stuck.
Not surprisingly, in the end of semester course evaluations my
students wrote about how much work they had done for this course. And yet, when
asked specifically whether or not they agreed with the statement, "The general
workload was appropriate for this level course," only five of my 33 students
disagreed. One student comment that summarized the tenor of the end of course
evaluations stated, "I would say that the course is difficult and a lot of
work, but very rewarding, because if you put in a lot of time and effort then
you can see yourself understand the material and do well. Although the course
can be really hard at times, there is always somewhere to go for help."
The second thing that happened this past week was my
discovery of How Learning Works: 7
Research-Based Principles for Smart Teaching [2]. This collaborative
effort, published in 2010, translates what has been learned by those engaged in
research in undergraduate education into practical guidance for those of us in
the classroom. What the authors call principles, I see more as facets of
teaching to which I need to pay attention. This is my own paraphrasing of these
principles or facets:
The need to understand the variety
of prior knowledge that my students bring to my class and how it helps or
hinders them.
The importance of how students
organize the knowledge they are acquiring and the need for me to understand
common misalignments and to help them make the necessary connections.
The critical role of student
motivation and my responsibility to strengthen it.
The need to develop automaticity
in basic skills and the fact that learning how to integrate and apply these
skills requires guidance and directed practice from me.
How important it is that I provide
useful feedback that is targeted at improving performance.
The role of the social, emotional,
and intellectual climate in my classroom.
The need for me to guide students in
practicing metacognition, monitoring what they are doing and why.
The book discusses the relevant research, but is also full
of examples of traps we can fall into and strategies for dealing with these
principles or facets in order to improve our teaching.
One trap discussed under #3 describes the teacher who, with
the intent of spurring his students to work hard, warned them at the start of
the course that they could expect that a third of them would not pass. This had
exactly the opposite effect. With the expectation that they would not do well
regardless of how much effort they put into the course, a large proportion of
the students directed their time and energy to other courses.
The issue here is motivation, getting the students to put in
the effort needed to learn the material. I believe that I did succeed
particularly well this past semester in motivating most of my multivariable
calculus students. How Learning Works identifies three levers that motivate students
to work hard. The first is value.
They have to believe that what I want them to learn will be of value to them.
Personal enthusiasm on my part goes a long way toward building this sense of
value. The second is a supportive
environment. They have to believe that the course is structured in such a
way as to help them be successful, rather than throwing up obstacles to their
success. Starting the projects and encouraging them to share their
understanding of homework problems within groups, providing feedback and
multiple opportunities to demonstrate understanding (as with WeBWorK and the chance
to earn back points lost on exams), and the availability of myself and my preceptor
build the sense of support. The third is self-efficacy,
belief that one is capable of achieving success.
This last is the main reason I will never again grade on a
curve. The message sent by grading on a curve is that the proportion of
failures has been determined in advance, regardless of how much work students
are prepared to invest in the course. It is also why I am disturbed that in our
national survey of calculus, faculty at the start of the term were able to
predict, almost perfectly, what their grade distributions would be at the end
of the term (see the last bullet under Instructor Attitudes in The Calculus I
Instructor, Launchings, June 2011). Going into this course, I would never
have predicted 67% A's. I am delighted that what I did worked so well with so
many of my students. [3]
Which brings me back to the issue of grade inflation. Grade
inflation is a red herring because it misdirects our attention from what should
be our true concerns: What do our grades mean in terms of expectation of
student achievement and understanding? And how can we support as many students
as possible to meet our highest expectations?
[1] Straffin, P. D., Jr. 1996. Hydro-Turbine Optimization. Pages 240–250 in Applications of Calculus. P.D. Straffin, Jr., editor. Classroom Resource Materials. MAA. [2] Ambrose, S. A., M. W. Bridges,
M. DiPietro, M. C. Lovett, M. K. Norman. 2010. How Learning Works: 7 Research-Based Principles for Smart Teaching.
Jossey-Bass. [3] Not all my students did well. The class GPA was 3.5. What was
important was that I had explicit expectations of what would constitute A work,
that I clearly communicated what was required to meet those expectations, that
students saw them as challenging but achievable, and that my students really
were graded according to these expectations.
Saturday, December 1, 2012
This past spring, the National Research Council of the
National Academies released its report, Discipline-Based Education Research: Understanding and Improving
Learning in Undergraduate Science and Engineering[1]. The charge to the
committee writing this report was to synthesize existing research on teaching
and learning in the sciences, to report on the effect of this research, and to identify
future directions for this research. The project has its roots in two 2008
workshops on promising practices in undergraduate science, technology,
engineering, and mathematics education.
Unfortunately, between 2008 and 2012 undergraduate mathematics
education dropped out of the picture. The resulting report discusses
undergraduate education research only for physics, chemistry, engineering,
biology, the geosciences, and astronomy. Nevertheless, it is an interesting
report with useful information—especially the instructional strategies that
have been shown to be effective—that is relevant for those of us who teach
undergraduate mathematics.
The studies that are described are founded on the assumption
that students must build their own understanding of the discipline by applying
its methods and principles, and this is best accomplished within a
student-centered approach that puts less emphasis on simple transmission of
factual information and more on student engagement with conceptual
understanding, including active learning in the classroom.
The great strength of this report is the wealth of resources
that it references and the common themes that emerge across all of the
scientific disciplines. A lot of attention
is paid to the power of interactive lectures. Given that most science and
mathematics instruction is still given in traditional lecture settings, finding
ways of engaging students and getting them to think about the mathematics while they are in class is essential
for increasing student understanding.
The recommendations of effective practice range from simple
techniques, such as starting each class with a challenging question for
students to keep in mind, to transformative practices such as collaborative
learning. A common intermediate practice involves student engagement by posing
a challenging question, having students interact with their peers to think
through the answer, and then testing the answer. In some respects, this is more
easily done in the sciences where student predictions can be verified or
falsified experimentally. Yet it is also a very effective tool in mathematics
education where a well-chosen example can falsify an invalid expectation and
careful analysis can support correct understanding. But most important is that it forces to try to
use what they have been learning.
In large classes, this type of peer instruction can be
facilitated by the use of clickers. The report does include the caveat, with
supporting research, that merely using clickers without attention to how they
are used is of no measurable benefit.
The greatest learning gains that have been documented occur
when collaborative research is incorporated into the classroom. The NRC report
includes many descriptions of how this can be accomplished in a variety of
scientific disciplines. It also references the research that has established
its effectiveness. Again, attention to how it is done is an important component
of effective practice.
Two of the areas that are identified as needing more
research are issues of transference (see my September column on Teaching and
Learning for Transference) and metacognition. Usefully, the authors point
out that there are two sides to transference: the ability to draw on prior
knowledge and the ability to carry what is currently being learned to future
situations. Metacognition is an important issue in research in undergraduate
mathematics education, especially for those studying the difference between
experts and novices engaged in activities such as constructing proofs. Experts
monitor their assumptions and progress and are prepared to change track when a particular
approach is not fruitful. Novices are more likely to choose what to them seems
the likeliest approach and then ignore alternatives.
In sum, this is a useful and thought-provoking report. I
wish that it had included undergraduate mathematics education research, but
perhaps that omission can be corrected as we move forward.
Thursday, November 1, 2012
One of the goals of the MAA Calculus Study, Characteristics
of Successful Programs in College Calculus, was to gather information about the instructors of mainstream
Calculus I. Here, stratified by type of institution, is some of what we have
learned, refining some of the data presented in "The Calculus I
Instructor" (Launchings, June 2011).
Again, I am using Research University as code for institutions for which the
highest mathematics degree that is offered is the PhD, Masters University if the
highest degree is a Master's, Undergraduate College if it is a Bachelor's, and
Two Year College if it is an Associate's degree. These surveys were completed
by 360 instructors at research universities, 73 at masters universities, 118 at
undergraduate colleges, and 112 at two year colleges.
Calculus I instructors are predominantly white and male.
Masters universities have the largest percentage of Black instructors, research
universities of Asian instructors, and two-year colleges of Hispanic
instructors. By and large, undergraduate colleges do not do well in
representing any of these groups.
There is a dramatic difference between the status and
highest degree of Calculus I instructors at research universities and those at
other types of colleges and universities. At research universities, instructors
are less likely to be tenured or on tenure track, or to hold a PhD. They are
also less likely to want to teach calculus: One in five has no interest or only
a mild interest in teaching calculus. The high number of part-time faculty at
masters universities and two year colleges is troubling because of the evidence
that such instructors tend to be less effective in the classroom and much less
accessible to their students [1]. Not surprisingly, less than a quarter of the
Calculus I instructors at two-year colleges hold a PhD.
Generally, calculus instructors consider themselves to be somewhat
traditional in their instructional approaches, and they believe that students
learn best from lectures. The greatest divergence from these views is at
undergraduate colleges where almost half consider themselves to be innovative
and 45% disagree that lectures are the best way to teach. The greatest variation
among faculty at different types of institutions is over the use of calculators
on exams. Close to half of the instructors at research universities do not
allow them; 71% of the instructors at two year colleges do.
There also are institutional differences in beliefs about
whether all of the students who enter Calculus I are capable of learning this
material. |
Success in your calculus course starts here! James Stewart's CALCULUS: EARLY TRANSCENDENTALS texts are world-wide best-sellers for a reason: they are clear, accurate, and fil [more].[less] |
Math Placement Test - Review Material
The following is a list of resources you may want to use to prepare for the mathematics placement test. The first three sections of the test are algebraic in nature; the fourth section deals with topics most commonly covered in a precalculus course. Any algebra text and/or precalculus text should contain the appropriate topics and provide enough review material for the placement test, so it is NOT necessary for you to study these particular texts. These are simply suggestions. |
Specification
Aims
The course unit will deepen and extend students' knowledge and understanding of algebra.
By the end of the course unit the student will have learned more about familiar
mathematical objects, will have acquired various computational and algebraic
skills, will have seen how the introduction of structural ideas leads
to the solution of mathematical problems and will have built a solid
foundation for any further study of algebra and algebraic structures.
Brief Description of the unit
Polynomials are familiar mathematical objects which play a part in virtually every branch of the
subject, e.g. solution of quadratic equations, the use of quadratic forms
to study maxima and minima of functions, approximation of functions by
Taylor or Chebyshev polynomials, the characteristic polynomial of a matrix,
etc. In algebra you have probably met problems involving factoring polynomials
and finding the gcd of two polynomials (in one variable) using the Euclidean
algorithm.
Historically the study of
solutions of polynomial equations (algebraic geometry) and the study of
symmetries of polynomials (invariant theory) were a major source of
inspiration for the vast expansion of algebra in the 19th and 20th centuries.
In this course the algebra of polynomials in n variables over a field of
coefficients is the basic object of study. The course covers recent advances
in the subject with important applications to computer algebra, together
with a selection of more classical material.
Learning Outcomes
On successful completion of this course unit students will be able to demonstrate
facility in dealing with polynomials (in one and more variables);
understanding of some basic ideal structure of polynomial rings;
ability to compute generating sets and Gröbner bases for such ideals;
ability to relate work with polynomials to the context of rings, ideals, and other
algebraic structures;
in particular, ability to solve problems relating to the factorisation of polynomials,
irreducible polynomials and extension fields;
facility in dealing with symmetric and alternating polynomials;
ability to relate work with symmetric functions to the context of invariants of finite
groups.
Future topics requiring this course unit
None.
Syllabus
Computing with Polynomials:
Polynomials in two or more variables, ideals in polynomial rings,
monomial
ideals, orderings on monomials, reduction and remainders, Gröbner
bases, Hilbert's basis theorem, S-polynomials and Buchberger's
algorithm.
Teaching and learning methods
Two lectures and one examples class each weekts. In addition students should expect to spend at least four hours each
week on private study for this course unit.
Course notes will be provided, as well as examples sheets and solutions. The notes will be
concise and will need to be supplemented by your own notes taken in lectures, particularly
of worked examples. |
2013-05-20T12:48:37ZIs Maths everywhere? Our students respond
Título: Is Maths everywhere? Our students respond
Autor/es: Mulero González, Julio; Segura Abad, Lorena; Sepulcre Martínez, Juan Matías
Resumen: Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe. In fact, Galileo said that Mathematics is the language with which God has written the universe. Aristotle defined mathematics as "the science of quantity", i.e., "the science of the things that can be counted". Now you can think that counting has a vital role in our daily life; just imagine that there were no mathematics at all, how would it be possible for us to count days, months and years? Unfortunately, people usually ignore the connection between mathematics and the daily life. Most of university degrees require mathematics. Students who choose not to take seriously mathematics or to ignore it in high school, find several difficulties when they come up against them at the university. This study explores the perceptions of how mathematics influences our daily life among our students and how teachers can use this information in order to improve the academic performance. The used research instrument was a questionnaire that was designed to identify their understanding on learning mathematics.2013-03-01T00:00:00ZDiseTítulo: DiseAutor/es: Marcilla Gomis, Antonio; Gómez Siurana, Amparo; García Cortés, Ángela Nuria; Beltrán Rico, Maribel; Reyes Labarta, Juan Antonio; Olaya López, María del Mar; Serra Ruiz, Marta
Resumen: Se presenta de forma resumida un ejemplo de desarrollo de materiales didácticos interactivos vía web (Marcilla y col. Ingeniería Química, 438, 153-160, 2006), en los que la utilización de las Tecnologías de la Información y la Comunicación (TIC) permite un nuevo tipo de herramienta docente en la que cada alumno puede elegir su propia ruta y velocidad de aprendizaje, pudiendo suponer tanto un refuerzo a las clases convencionales como una posible vía para el autoaprendizaje y docencia "no presencial". [
Descripción: Presentación realizada para las II Jornadas de Intercambio de Ideas
entre Docentes de Titulaciones de Ciencias de Universidades Valencianas,
Vicerrectorado de Calidad y Armonización Europea UA (Alicante), Junio 2007.2007-06-01T00:00:00ZFundamentals of Physics in Engineering I: course in OCW-UA (academic year 2012-2013)
Título: Fundamentals of Physics in Engineering I: course in OCW-UA (academic year 2012-2013)
Autor/es: Beléndez Vázquez, Augusto
Resumen: Course published in the OpenCourseWare-UA corresponding to the subject "Fundamentals of Physics in Engineering I" that is taught in the first year of the "Degree in Sound and Image, in Telecommunications" of the Polytechnic School at the University of Alicante. This course includes topics guides, summaries and proposed problems.2012-12-12T00:00:00ZFundamentals of Physics in Engineering I: summaries of the units (Academic year 2011-2012)
Título: Fundamentals of Physics in Engineering I: summaries of the units (Academic year 2011-2012)
Autor/es: Beléndez Vázquez, Augusto
Resumen: Summaries of the units of course "Physical Foundations of Engineering I". Degree in Sound and Image Engineering, in Telecommunications. Polytechnic School of the University of Alicante.2012-03-15T00:00:00Z |
This course is an introduction to the basic ideas of ordinary differential equations. Topics include linear differential equations, series solutions, simple non-linear equations, systems of differential equations, and applications.
Expected Educational Results
As a result of completing this course, the student will be able to do the following:
1. Analyze problems using critical thinking skills.
2. Use functions and their derivatives to construct mathematical models.
3. Solve application problems for which differential equations are mathematical
4. Solve the following kinds of first order, ordinary differential equations:
a. separable
b. homogeneous
c. exact
d. linear, and
e. Bernoulli.
5. Solve second order linear ordinary differential equations:
a. Homogeneous and non-homogeneous equations with constant coefficient
b. Power series solutions about ordinary and regular singular
6. Solve initial value problems using Laplace transforms.
7. Solve systems of linear differential equations.
8. Approximate a solution to a differential equation with a numerical method
9. Use some basic commands of a computer algebra system, and solve differential equations with them.
10. Determine the stability of linear systems.
11. Analyze almost linear systems.
12. Use the Energy Method to describe nonlinear systems.
13. Be able to identify the basic forms of bifurcation.
General Education Outcomes
I. This course addresses the general education outcome relating to communication by additional support as follows:
A. Students develop their listening and speaking skills through participation and through group problem solving.
B. Students develop their reading comprehension skills by reading the text and the instructions for text exercises, problems on tests, or on projects. Reading mathematics text requires recognizing symbolic notation as well as problems written in prose.
C. Students develop their writing skills through the use of problems requiring written explanations of concepts.
I. This course addresses the general education outcome of demonstrating effective individual and group problem-solving and critical thinking skills as follows:
A. Students must apply mathematical concepts previously mastered to new problems and situations.
B. In applications, students must analyze problems and describe problems with their pictures, or diagrams, or graphs, then determine the appropriate strategy for solving the problem.
I. This course addresses the general education outcome of using mathematical concepts to interpret, understand, and communicate quantitative data as follows:
A. Students must demonstrate proficiency in problem-solving skills including applications of differential equations and systems of differential equations.
B. Students must write differential equations to describe real-world situations and interpret information from the solution of differential equations and systems of differential equations.
C. Students must solve equations and systems of equations (both linear and which often arise in modeling numerical relationships.
I. COURSE GRADE
Exams, assignments, and final exam prepared by individual instructors will be used to determine the course grade.
II. DEPARTMENTAL ASSESSMENT
This course will be assessed every three years. The assessment instrument will consist of a set of open-ended questions, which will be included as portion of the final exam for all students taking the course.
A committee appointed by the Academic Group will grade the assessment material.
III. USE OF ASSESSMENT FINDINGS
The MATH 2652 committee, or a special assessment committee appointed by the Academic Group will analyze the results of the assessment and determine implications for curriculum changes. The committee will prepare a report for the Academic Group summarizing its finding.
Effective date of offering: Summer 2002
CCO completed 12/06/01
Reviewed by committee April 2005 |
In this differentiation instructional activity, students select from a list of differentiation rules which one they have to use to find the derivative of the given equation. They evaluate one integral exactly.
Directions are written to solve a related rate problem step by step. There are five example problems to practice solving for related rates. Use of the Chain Rule and/or implicit differentiation is one of the key steps to solving these word problems.
Students review vocabulary words for calculus. In this calculus lesson, a list of vocabulary words is provided for students to review. Students may use this list as a review of important terms to know for calculus.
Young scholars calculate the velocity of object as they land or take off. In this calculus lesson, students are taught how to find the velocity based on the derivative. They graph a picture the represent the scenario and solve for the velocity. |
Little Silver Trigonometry home, the predominant language spoken is mandarin. Taken 2 years of AP Calculus AB and BC. Covers differential, integral calculus, and its applications.Calculus is a science of interpreting formulas
and analyzing different curves. Usually curve is defined by area under its graph, named integral and velocity (speed or rate in concrete moment),named differential.To imagine
graph we need to know how it is behave on the ends 0 and infinity. For th... |
Bringing a new vitality to college mathematics
Modularized mathematics is a common curricular strategy in our era, with a common justification and design strategy being the identification of what math students need. Separately, I have posted about the use of modules (and I will have more to say on them); today, this is about the use of 'identifying the math they need'.
Here is a short story, a parable, with your indulgence:
Felicia and Ashley have been managing a service-oriented hardware store in their town for five years, and they finally have enough capital accumulated to remodel their store. In their planning process, they realize that it is important to make sure that they effectively meet the needs of their customers. With the help of a PR company, Felicia and Ashley design a web survey form that the customers can use to identify the items and categories of need. Naturally, the items and categories are based on what the store has already been selling. Many customers complete the survey, with a surprising consistency in the general results. Based on the results of the survey, a remodeled store opens with the merchandise reflecting the survey … items needed by many are in-stock and visible in an attractive display; items needed by a few are done as a special order.
After two months, it becomes clear that the new store is far less profitable than the old. A new survey is done to determine the problem, including areas for general comments. The results of this survey show that there were two causes of the problem. First, it turns out that the 'items needed by a few' were significant as a group … many items "less needed" accumulated over many customers creates a large change; the special order process did not meet the needs. Second, and mentioned on every comment, is the fact that there were four areas of emerging need in hardware that were not listed on the original survey; since they were not even listed, customers could not report this need. These emerging needs reflect both the newest do-it-yourself projects and the maintenance of the newest homes.
When we design modules or courses based on a content survey, we are beginning with the assumption that "what is needed" is within the existing content. This survey approach is commonly used for module designs, as well as research on the mathematics needed in various occupations. If we run a hardware store, there is an implied responsibility to understand 'hardware-ology' deeply to understand the needs of the patron even before they know what they need.
We run a mathematics learning enterprise. We carry a responsibility to deeply understand the mathematical needs of our students. Our situation is, in fact, far worse than the hardware store in the parable; the hardware store was successful before the change to 'needed' items. Mathematics programs …developmental or college credit … are definitely not successful currently. If a particular math program was already successful, there would not be much motivation to 'modularize' or to identify math needs; the fact that a program is modularized is a direct statement of non-success.
I suggest that you consider the more basic question:
Is it possible that our mathematics programs are not meeting the mathematical needs of our students and that this is a major factor in the program not successful?
Existing developmental math content is based on an archaic set of school mathematics content; it does not reflect the changes in schools since 1965. Existing introductory college mathematics is based on a curriculum extended from that archaic foundation. As you know, extrapolating from a model to a new set of domain is a risky process; not only do we assume the validity of the original model (not justified in math), we assume that the extrapolation is valid. We have significant curricular studies that conclude that the extrapolation is not well founded; see the work of MAA 'CRAFTY' (
Here is what we need instead of 'need based on current content':
We need to identify the basic mathematical knowledge needed for our students to be prepared well for the mathematical needs of their college academic work as well as societal needs.
A friend of mine is a somewhat famous economics educator in community colleges. Current economics work is very advanced mathematically; however, introductory economics (micro, macro) are taught qualitatively with very small doses of quantitative work. The reason? It's not that economics educators don't want or need quantitative methods at the introductory level … the reason is that their students are woefully prepared for quantitative work, even after algebra courses. We 'cover' slope, but not rate of change in general (for math courses most students take); what we do cover is done in a way that inhibits transfer of learning to a new setting (economics). I've had this same conversation with science faculty, with the same result; I expect that much of the same story would be found in some social sciences.
The use of modules in curricular design raises issues about learning mathematics. The use of 'what students need', when based on existing content, reinforces an archaic model of mathematics. It is our responsibility to understand our students' mathematical needs at a deep level, to the depth that we can identify content that is outside of the current curricula. If we can not judge this need, nobody else will.
The New Life model was based on exactly this type of work; we identified the needs based on a professional understanding of the quantitative demands of current students, especially those in community colleges. Some of this work is now imbedded in the Carnegie Pathways, and has a similar development in the Dana Center's "New Mathways Project". The curricular design in these efforts seeks to begin meeting students quantitative needs, starting on the first day of their first math course (developmental or not).
"For us [Bricmont and Sokal], the scientific method is not radically different from the rational attitude in everyday life or in other domains of human knowledge. Historians, detectives an plumbers—indeed, all human beings—use the same basic methods of induction, deduction and assessment of evidence as do physicists or biochemists."
(Intellectual Impostures)
Actually, mathematics is a related set of ways of thinking. The issue is which of these are most helpful to students. Some of us think that the best answer is a type of 'quantitative reasoning' in place of the typical developmental algebra work. |
This course will emphasize the study of linear functions. Student will use functions to represent, model, analyze, and interpret relationships in problem situations. Topics include graphing, solving equations and inequalities, and systems of linear equations. Quadratic and nonlinear functions will be introduced. This course counts for high school credit and will become a permanent part of the student�s high school transcript. Grade 8 Algebra I will be factored into the student�s overall high school grade average/GPA. Students will be administered the 2012 STAAR EOC for Algebra I in May. Students EOC score will NOT count as 15% of their final Algebra I grade. In order to graduate from high school, students must achieve a cumulative score in each of the four core content areas. For each of the four core content areas, the student�s cumulative score ≥ n times student�s passing scale score, where n = number of assessments taken. For more information regarding the STAAR EOC go to the TEA website at
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worksheet will allow you to visually see how slope changes using dynamic text and Sliders. By using Sliders students can see how the slope or the steepness of the line changes with respect to the values of x and y. Students will al
Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques. In F.LE Equal Differences over Equal Intervals 2, students prove the property in general (for equal intervals of any length).
In this task students observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
In this task students have the opportunity to construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
This lesson is designed for students to gather and analyze data about baseball figures. The student will use the Internet or other resources to collect statistical data on the top five home run hitters for the current season as well as their career home run totals. The students will graph the data and determine if it is linear or non-linear. |
Real Number System
Dr. Eaton will cover the Real Number System and all its subcategories such as natural numbers, whole numbers, integers, and rational numbers. You will also cover terminating and repeating decimals as well as square roots, irrational numbers, and perfect squares. The lecture rounds out with the number line and an additional four examples to make sure of your understanding.
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Real Number System
Important subsets
of the real numbers are: the natural numbers, the whole numbers, the
integers, the rationals, and the irrationals.
The number line
can be used to graph sets of numbers.
Each positive
number has 2 square roots. The positive one is called the principal
square root.
Use decimal
approximations of irrational numbers to compare and order a set of
real numbers that includes some irrationals.
Real Number System
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
The lack of some explanations and the bad explanations of some of the
key concepts in the first chapter are disappointing. I want to try to fill
in the gaps in order to make understanding and doing algebra easier.(1)
First, the book makes the knowing the names of some the principles
and some of the properties discussed seem more important than understanding
the concepts, whereas it is understanding the concepts that is important.
In some cases, the names the author gives the principles or the properties
are arbitrary and non-descriptive; and in some cases they are very misleading.
The eighth grade chapter 1 pre-test had at least one question in which
knowing the names of some principle was necessary in order to get the answer
right, even though knowing the name is not especially important for being
able to do algebra. Knowing the name of the principles is neither necessary
nor sufficient for understanding the principles; and many kids will know
the names and even be able to work some problems, but only paradigmatic
ones, without knowing what they are doing or why it works. I think this
is not good. It is much more important that students understand the concepts,
properties, relationships, and principles involved than that they know
names of these things, particularly when knowing the names does not show
or mean understanding. If grades are to be given, it should be on understanding
the items, not naming them, particularly when the names used in the text
are arbitrary.
The reason understanding the concepts is important in algebra is that
math is generally about the logical relationships of certain things, such
as numbers, and the logical properties these things have. In algebra,
you usually have to use just (relevant) logical properties and (relevant)
logical relationships, along with a few given facts, in order to derive
some things you need to know but are not given or told.Basically
all you have to work with is your understanding of number properties and
relationships. And it is your understanding properties and relationships,
not your naming them, that is important for doing algebra. And the better
your understanding of logical properties and logical relationships, the
better chance you have of figuring out which ones are relevant to a given
problem, though knowing what is relevant often takes luck or a flash of
insight(2). Naming only allows an abbreviation
for discussing properties and relationships with someone else --provided
they know the same names you do.
With regard to what the book calls the distributive principle (of multiplication):
(1) that principle involves two principles actually: a) what my teachers
called "multiplying (an expression) out" an expression of the
form a(b+c), and (b) "factoring" or finding factors for an expression
of the form ab+ac. For example, not only is it important to know
a(b+c) equals ab+ac, [equation 1]
but it is just as important to know this in the "opposite direction",
that
ab+ac equals a(b+c). [equation 2]
This is because in some algebra problems or steps it will be important
to "multiply out" the expression a(b+c) in order to get ab+ac;
but in other problems or steps it will be important to change expressions
into their "factors", so that, in this case, you would want to
change ab+ac into a(b+c). The reason it will be important to go one way
rather than the other is that one form of the equation will be more useful
than the other in some cases. For example, if you have to solve: 5(3x -
11) = 7x - 3(x+10), multiplying it out is important in order to find out
how many total x's you are dealing with, and what total purely numerical
quantity you are dealing with. However: in a problem like (9x2
- 9y2) ÷ 3(x-y), it will be easier to solve the problem
by factoring the first expression into 9(x2-y2) and
then factoring (x2-y2) into the two factors which
are (x+y)(x-y), as you will learn later in algebra.(3)
This gives 9(x+y)(x-y)÷3(x-y), which is easier to see gives 3(x+y).
(4)
(2) The book does not explain why distributive multiplication works;
and most students don't understand how it works. Enclosed is a diagram
that shows how it works. It is a diagram
of rows and columns showing why (a+b)(c+d)= ac+ad+bc+bd. You could
make up a similar diagram for yourself of a(b+c). Or suppose a=8, b=3,
and c=4. Then, according to the principle of distribution (or of "multiplying
out"), 8(4+3) should be the same as (8x4)+(8x3). Since 32+24 = 56,
you can see it works in that case. But also think of it as laying out 8
rows and 7 columns of something, such as chocolate chips. No matter how
you divide up the rows and columns with partitions [for example, 8(3+4)
or 8(6+1) or 8(2+3+2) or (5+3)(2+3+1+1)], the total number of chips will
not change; and the subgroups will all be little rectangles of (sub-rows
x sub-columns), which, when added up will be the same number of chips as
the total number of rows times the total number of columns. You may understand
the principle in some other way, but you should figure out for yourself
some way to see why or how the principle works. It should make sense to
you. Then you don't have to try to remember it or worry about getting it
confused with some other equation that is not universal.
(3) The distributive principle is what is behind the way we multiply
"by hand", to get the product of numbers such as 35x82.
35 x 82 70 280 2870
If you were taught to multiply in the above way, multiplying 35 by 2
and then 35 by 80 (or 8, but starting the answer in the "tens column")
you are essentially multiplying 35(80+2) and getting (35)(80)+(35)(2).
That is why you end up adding together the two numbers 70 and 2800 after
you multiply the parts separately. [Actually, if you want to get precise,
you are multiplying (30+5)(80+2) to get (30x80)+(30x2)+(5x80)+(5x2), only
in a slightly different order, but it all comes out the same.]
Reflexive, Symmetrical, and Transitive Properties
This is not explained well at all in the book. "Equality"
or "being equal" is one of many relationships which are logically
reflexive, symmetrical, and transitive. There are many such properties
in the universe. Being "the same height as" is such a property,
for (1) everyone is the same height as him/herself (reflexive), (2) everyone
is the same height as anyone the same height as him (symmetrical), and
(3) everyone is the same height as a third person who is the same height
as someone else who is the same height as the original person (transitive).
The last one of these is a little hard to follow, but it essentially says
something like, if you are the same height as Frank, and Frank is the same
height as Sally, then you are the same height as Sally.
A property or relationship is reflexive if and only if everything
has that property in relationship to itself. Being a grandfather is
not a reflexive property because not everyone is their own grandfather
(actually no one is, of course). Being "as smart as" is reflexive,
since everyone is as smart as him/herself.
A property or relationship is symmetrical if and only if in every
case where one thing has that relationship to a second thing, the second
thing logically has that relationship to the first thing. Being "a
sister to" is not reflexive, since Sue can be a sister to George,
but that does not mean George is a sister to Sue. Being "a sister
to a female" IS reflexive, however, isn't it?
A property or relationship is transitive if and only if in every
case, whenever one thing has that relationship to a second thing, and the
second thing has that relationship to a third thing, the first thing logically
has to have that relationship to the third thing also. Being "a
cousin of" is not transitive because not all the cousins of your cousins
are cousins to you. For example, you are the cousin of both your father's
brother's kids and your mother's sister's kids, but they are not cousins
to each other. Being "taller than" is transitive because if you
are taller than Paul, and Paul is taller than John, you must be taller
than John also.
Being "equal to" is reflexive, symmetrical, and transitive
since everything is equal to itself (reflexive), equal to whatever is equal
to it (symmetrical), and equal to anything equal to anything equal to it
(transitive). The last one is, of course, hard to understand put into those
words, but it essentially says that if A is equal to B, and B is equal
to C, then A must also be equal to C.
The reason any of this is even said about "equality" is that
when you are trying to solve problems in algebra, often you will use these
properties of equality to make substitutions for equivalent formulas, quantities,
or expressions in order to do calculations more easily. It is just that
you will do it without thinking of the names "reflexive", "symmetrical",
and "transitive". For example, if you have
(3a+3b)÷(b+a),
you can substitute 3(a+b) for (3a+3b), since they are equal, and then
substitute (a+b) for (b+a) because they are equal, and that makes it easy
to see how to divide, and get the answer, 3, since the problem is the same
as
3(a+b)÷(a+b).
And substituting (a+b) for (b+a) also shows why it is important to keep
in mind that addition and multiplication can each be done in any order
separately ("the commutative property") though not mixed
up together, since by arbitrary, but binding, convention (unless it
is changed) 3+6x8 [51] is not the same as 6+3x8 [30]. Similarly, there
may be times of regrouping or re-associating addends or multiplicands
in order to see relationships more clearly. I cannot think of such a case,
but perhaps you will come across one some day.
What the book calls "the additive and multiplicative identity properties"
are, of course, just the facts that anything zero is added to does not
change, and anything multiplied by 1 does not change. It is certainly easier
to remember these ideas than the names the author gives them. They should
be able to be expressed clearly in terms other than the names the author
has mentioned.
Order of Operations
One of the intial difficulties students often have is understanding
the rationale for "order of operations"; that is, which calculations you do in what order when you calculate a sequence of numbers separated by signs, or having exponents, etc.; for example, the principle that multiplication
and division take precedence over addition and subtraction, so that in a
sequence like the following:
6 + 9 divided by 3
the answer should be 9 rather than 5. It is not because of some logical
or mathematical relationship that the answer should be 9 rather than 5.
It is because the convention of the meaning of the way the numbers
are written is that you group the numbers connected by multiplication
and division first, and perform those operations, before you then add or
subtract the quantities connected by plus or minus signs. So that by convention
(not by logic) the above sequence really means 6
+ (9 divided by 3), instead of (6 + 9) divided
by 3.
6 + (9 divided by 3) is 6 + 3, which is
9
whereas (6 + 9) divided by 3 is 15 divided
by 3, which is 5.
Now the trick in understanding "order of operations" is realizing
that there is nothing to "understand", since "order of operations"
is not a matter of logic, but a matter of convention. That is, it
is simply an edict that IF numbers are written without groupings
by parentheses, THEN the order one is supposed to deal with them
is by the arbitrary, but agreed upon, rules of "order of operations".
This is a convention simply because it is the way that was chosen
even though other ways could have been chosen that would have been
different and might have worked just as well. They would have worked just
as well as long as everyone knew what the rules were. Driving on
the right side of the road in the United States, and on the left side of
the road in England are examples of conventions. It doesn't matter which
side of the road everyone drives on, as long as a side is chosen and everyone
THEN knows what it is and honors it. Similarly it is a convention
that English is written and read from left to right and from top to bottom
rather than from bottom to top or from right to left. The reason for written
language conventions, which is what "order of operations" is,
is so that people can read what others (or themselves later) have written.
In math, order of operations tells you how to read what I have written
if I happen to leave out the parentheses and write:
3 times 6 + 42 divided by 7 + 3squared
You will know what I meant, and I will also know what I meant when I
come across my own notes later. Without a convention of some sort about
this kind of thing, it would be difficult to communicate with each other
about mathematical sequences, for you might easily perform the operations
in a different order from what I was trying to tell you to do. It is normally, however, far safer to use parentheses
and brackets, etc., to group quantities than to rely on being sure you
have remembered "order of operations" correctly when you write
down your sequences, and when you go to calculate them.
1. Some of the explanations and mathematical proofs I give will be clear,
I think, if you patiently work your way through them. Unfortunately,
these things are hard to say in a way that makes them clear without your
having to think about them and work through them yourself. I try to show
a way to work through them, and though this way makes sense to me,
you have to make it-- or your own way of thinking about these things--
make sense to you. Someone else cannot automatically make such complex
things make sense to you no matter how they say them; only you can make
sense out of complex explanations --by thinking about them.
2. When Richard Feynman, a gifted mathematician and
physicist, was on his high school math team, one of the problems he solved
in a contest long before everyone else was this one: Suppose you and a
crew of friends are rowing upstream at the rate of 7 miles per hour relative
to the shore and that the current is flowing at the rate of 2 miles per
hour, relative to the shore. Your hat falls into the water and starts floating
downstream fifteen minutes before you notice it. How long would it take
you to catch up to the hat if you instantaneously reversed direction? Feynman
immediately saw that the answer was 15 minutes because he saw that the
rates given in relation to the shore were irrelevant, just like when you
leave something at home by accident and drive 15 minutes before you realize
it. It will take you 15 minutes at the same speed to get back to it, even
though the earth has moved relative to the sun and has also been rotating
at the rate of 1000 miles per hour in perhaps the opposite direction you
were driving. I still have difficulty seeing this with regard to
the water case, but Feynman saw it right away.
Often it is easy to focus on the "wrong" or least helpful
relationships. There is a famous trick problem that is not difficult to
solve if you don't know calculus, but is difficult to solve if you do know
calculus. The reason is that the easier solution does not use calculus,
but the problem is set up to look like the sort that it takes calculus
to solve. That problem is: Two trains start out 1500 miles apart on the
same track, going straight toward each other. One train is moving 40 miles
per hour and the other is going 35 miles per hour. At the exact moment
they start, a bee that can fly 100 miles per hour starts from one and flies
to the other, instantaneously reversing direction when it gets to the other
train. The bee does this over and over --flying shorter and shorter distances
each time between the trains as they travel toward each other-- until it
is crushed between the two trains when they crash into each other. How
far in total distance will the bee have flown. To do this problem using
calculus requires what is called "summing an infinite series",
which is a complex calculation. But in arithmetic, it is easy: the trains
are headed toward each other at the combined rate of 75 miles per hour.
Thus it will take them 20 hours to cover the 1500 miles. Since the bee
is flying 100 miles per hour for that 20 hours, the bee flies 2000 miles.
It is not always easy to know which logical relationships are the most
productive to use when solving a math problem.
and which is x2-y2. In the given problem, however,
reversing this procedure to end up with the factors (x+y)(x-y) is the more
important direction.
By the way, the formula or equation (x+y)(x-y)=(x2-y2)
comes in handy in multiplying some numbers in your head. For example, 48
x 52 is the same as (50-2)(50+2), and since this is equal to 502
- 22, the answer is fairly easily computed to be 2496, since
that is 2500 - 4. You can do this in multiplying together any two numbers
equidistant from some easily squared number. Appearing to do such multiplications"in your head" will impress some people; don't tell them the
trick.
4. There is a famous fake "proof" that 2=1, using some of
these principles:
Let x=y
Then multiplying both sides by x, we get
x2 = xy
Then subtract y2 from both sides, giving
x2 -y2 =xy-y2
Factoring both sides, we get
(x+y)(x-y)=y(x-y)
Dividing both sides by (x-y) gives
x+y = y
Since x=y and therefore y=x, we can substitute y for x, giving
y+y = y
which is 2y = y
and dividing both sides by y, gives
2 = 1
The "bad" step is dividing both sides by (x-y), since in
this case, where x and y are equal, that is dividing by 0.
A more interesting, real proof using some of these principles gives
a surprising, but actual result. Consider this problem:
Suppose you had a smooth ball the size of the earth, and tied a 24,000
mile long ribbon around its middle, so that it was tight against the ball.
Then add a piece of extra ribbon to it -- a piece that is just 1 yard (i.e.,
36 inches) long; and smooth the ribbon out all the way around, so that
the loop disappears and the ribbon now forms a circle again. Will the ribbon
be very far off the surface of the ball? How far? It seems like it would
hardly raise the ribbon at all; but actually it will raise the ribbon about
six inches off the ball all the way around. In fact, it will turn out that
not only is the circumference of every ball roughly six times its radius,
but increasing any circumference of any circle by some amount will increase
its radius by nearly one-sixth that amount also.
The circumference, C, of any circle is 2piR, where R is the radius of
the circle.
Dividing both sides of C=2piR by 2pi, we get the formula for the radius
in terms of the circumference: R=C/2pi
In the problem with the ball and the ribbon, what we are looking for
is the difference in the radius of the ribbon when it is 24,000 miles long
and when it is 24,000 miles and 36 inches long. That will give us the distance
the ribbon is away from the ball.
If we use subscripts to designate the first and second radius and the
first and second circumference, the new radius will be R2 =
C2/2pi
and since the new circumference is 36 inches longer than the first
circumference, we can substitute (C1+36") for C2
to get: R2 = (C1+36")/2pi
Since a/x + b/x = (a+b)/x, (a+b)/x = a/x + b/x,
And by this principle we can rewrite our result to get: R2
= (C1/2pi)+(36"/2pi)
Since (C1/2pi) is the old radius, this means R2 =
R1+(36"/2pi)
And since 2pi is 6 when rounded off to a whole number, this means that
the new radius is the old radius plus approximately 6 inches. And, in general,
the new radius of any circle will be the old radius plus one-sixth of the
amount added to the circumference. So whether you add 36 inches to the
circumference of a dime or to the circumference of the earth, you increase
the radius of each of them by the same amount, roughly six inches. Hard
to believe, but true. |
How Do I Find Out How to Do Things in
Mathematica?
Mathematica comes with an extensive built-in help system to
assist you in learning how to best use the software. The Help Browser is a
complete online reference including all 1,470 pages of The Mathematica Book
and all supplemental documentation. A good place to start
learning about Mathematica is in the "Tour of Mathematica" found
in the Help Browser, in
the hard copy of The Mathematica Book, or on this web site. |
Rheems Calculus
...Just watching a teacher work through a problem does not necessarily mean the student can do the problem on his/her own. Thus, it is very important to practice problems on their own. In the classroom, I would give an example of a problem, and then give time for the students to work a similar problem on their own in class. |
Brand new. We distribute directly for the publisher. Winner of the CHOICE Outstanding Academic Book Award for 1997! The purpose of this book is to teach the basic principles [more]
Brand new. We distribute directly for the publisher. Winner of the CHOICE Outstanding Academic Book Award for 1997! The purpose of this book is to teach the basic principles of problem solving, including both mathematical and nonmathematical problems. This book will help students to...* translate verbal discussions into analytical data. * learn problem-solving methods for attacking collections of analytical questions or data. * build a personal arsenal of internalized problem-solving techniques and solutions. * become "armed problem solvers", ready to do battle with a variety of puzzles in different areas of life. Taking a direct and practical approach to the subject matter, Krantz's book stands apart from others like it in that it incorporates exercises throughout the text. After many solved problems are given, a "Challenge Problem" is presented. Additional problems are included for readers to tackle at the end of each chapter. There are more than 350 problems in all. This book won the CHOICE Outstanding Academic Book Award for 1997. A Solutions Manual to most end-of-chapter exercises is available.[less] |
Course
Number: MAM.ALG3
Course Name: Algebra 3 Prerequisite: None Course Description:
This course is designed to meet the NCTM standard. It is a formal study of algebra including applications. Students are expected to have mastered the content of Math 7 and Math 8. They use computers, calculators, and manipulatives in their study of the real number system, its properties and operations. This course will include solving first- and second-degree equations, operations on polynomials, and applications of algebraic methods of problem-solving. Course Length: 1
semesters Period Length: 1 Grade Level: 6-8
grade(s) Credit Per Semester:n/a |
complete, interactive, objective-based approach, "Intermediate Algebra: An Applied Approach," is a best-seller in this market. The Seventh Edition provides mathematically sound and comprehensive coverage of the topics considered essential in an intermediate algebra course. An Instructor''s Annotated Edition features a comprehensive selection of instructor support materials. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students interact with and master the concepts as they are presented. This approac... MOREh 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'' complete solutions help students identify their mistakes and preventfrustration."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 morethan 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
Chapters 2–6 are followed by Cumulative Review Exercises
Review of Real Numbers
Introduction to Real Numbers
Operations on Rational Numbers
Variable Expressions
Verbal Expressions and Variable Expressions
Focus on Problem Solving: Polya's Four-Step Process Projects and Group Activities: Water Displacement
Focus on Problem Solving: Another Look at Polya's Four-Step Process Projects and Group Activities: Solving Radical Equations with a Graphing Calculator, The Golden Rectangle
Quadratic Equations
Solving Quadratic Equations by Factoring or by Taking Square Roots
Solving Quadratic Equations by Completing the Square
Solving Quadratic Equations by Using the Quadratic Formula
Solving Equations that are Reducible to Quadratic Equations
Quadratic Inequalities and Rational Inequalities
Applications of Quadratic Equations
Focus on Problem Solving: Inductive and Deductive Reasoning Projects and Group Activities: Using a Graphing Calculator to Solve a Quadratic Equation
Functions and Relations
Properties of Quadratic Functions
Graphs of Functions
Algebra of Functions
One-to-One and Inverse Functions
Focus on Problem Solving: Proof in Mathematics Projects and Group Activities: Finding the Maximum or Minimum of a Function Using a Graphing Calculator, Business Applications of Maximum and Minimum Values of Quadratic Functions
Exponential and Logarithmic Functions
Exponential Functions
Introduction to Logarithms
Graphs of Logarithmic Functions
Solving Exponential and Logarithmic Equations
Applications of Exponential and Logarithmic Functions
Focus on Problem Solving: Proof by Contradiction Projects and Group Activities: Solving Exponential and Logarithmic Equations Using a Graphing Calculator, Credit Reports and FICOreg; Scores
Conic Sections
The Parabola
The Circle
The Ellipse and the Hyperbola
Solving Nonlinear Systems of Equations
Quadratic Inequalities and Systems of Inequalities
Focus on Problem Solving: Using a Variety of Problem-Solving Techniques Projects and Group Activities: The Eccentricity and Foci of an Ellipse, Graphing Conic Sections Using a Graphing Calculator |
Mathematics: Title III Grant
Title III Grant: Mathematics Foundation and STEM Success
The Title III Mathematics Foundation and STEM Success grant funds a five-year project designed to improve student success in mathematics and establish the quantitative fluency essential for success in science, technology, and engineering courses. The project began in October 2008 and runs through September 2013 and directly affects all faculty members in the STEM disciplines: science, technology, engineering, and math.
Central to the project is the mapping and re-design of the core mathematics sequence. From developmental to advanced math, course competencies are mapped to identify gaps, inconsistencies, and unnecessary redundancies. The result is curriculum realignment and courses are being re-designed to incorporate the best research-based practices for student success.
In conjunction with pilot courses, the project is working to improve on-campus learning support services for mathematics including the Mathematic Department's "Math Assistance Center" (MAC) and Freshman Math Program's "Algebra Alcove," along with the class, Independent Studies in Mathematics (ISM), which offers individualized instruction to help students be successful in their math courses.
Because math competency plays a crucial role in student success in all STEM programs, the math faculty and Title III staff collaborate with faculty in other STEM disciplines to integrate key mathematical concepts into all STEM courses. The second year of the project (2009-2010) focused on Biology courses, the third year (2010 – 2011) on Chemistry courses, the fourth year (2011-2012) focuses on Geosciences, and the fifth year (2012 – 2013) on Engineering and Physics.
If you have questions about the Mathematics Foundation and STEM Success, please contact Project Director Rolly Constable: [email protected]; 970-247-7234
Independent Studies in Mathematics
Independent Studies in Mathematics (ISM) are classes designed to help Fort Lewis students develop essential math skills through individualized instruction so they are prepared for success in their College Math Courses. Whether a student is having difficulty in a math class, failing a course, or can't enroll in a needed math class, they can enroll in an ISM course and--with a study-plan based on their individual needs--develop math skills in Arithmetic, Geometry, Algebra, Trigonometry, or Pre-Calculus.
ISM is variable credit course--one, two, or three credits, and because it's a late–registration course, students can enroll during the semester--whether or not they are currently enrolled in a Math Class-- and get individual help to succeed in their current or future Math classes.
ISM has limited enrollment and to enroll students will need a referral from their advisor or math instructor. |
Hailed by The New York Times Book Review as "nothing less than a major contribution to the scientific culture of this world," thisTable of Contents for Mathematics: Its Content, Methods and Meaning
Volume 1. Part 1
Chapter 1. A general view of mathematics (A.D. Aleksandrov)
1. The characteristic features of mathematics
2. Arithmetic
3. Geometry
4. Arithmetic and geometry
5. The age of elementary mathematics
6. Mathematics of variable magnitudes
7. Contemporary mathematics
Suggested reading
Chapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii)
1. Introduction
2. Function
3. Limits
4. Continuous functions
5. Derivative
6. Rules for differentiation
7. Maximum and minimum; investigation of the graphs of functions
8. Increment and differential of a function
9. Taylor's formula
10. Integral
11. Indefinite integrals; the technique of integration
12. Functions of several variables
13. Generalizations of the concept of integral
14. Series
Suggested reading
Part 2.
Chapter 3. Analytic Geometry (B. N. Delone)
1. Introduction
2. Descartes' two fundamental concepts
3. Elementary problems
4. Discussion of curves represented by first- and second-degree equations |
Electrc 2011 performs many electrical contracting and engineering calculations in complete conformance with the 2011 National Electrical Code (NEC). It produces detailed professional printouts as well as on-screen details. Many NEC parameters.
The program performs visualization of 4 most popular graph algorithms: Dijkstra, Floyd, Prim and Kruskel algorithms. It supports definition of color and width of edges, color and size of vertices, and step delay timeCounting wheel is a simple to play game that practises the skills of number recognition, counting and hand/eye co-ordination. Its age neutral design means it is equally suitable for child and adult basic skills studentsStatistics Problem Solver is a tutorial software that can solve statistical problems and generate step-by-step solutions. Statistics help by solving your statistical problems and demonstrating the various steps and formulas that are involved.
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Improve basic academic English skills in reading, spelling and written expression via any subject materials. Enables you to easily program a PC with subject materials. Multimedia program produces on-screen exercises and printed worksheets.
Whole number and fraction computation skills. Number by number problem exercises. Includes whole number math facts, addition, subtraction, multiplication and division of whole numbers and fractions. Also includes English and Metric measurements.
Whole number and fraction computation skills. Number by number problem exercises. Includes whole number math facts, addition, subtraction, multiplication and division of whole numbers and fractions. Also includes English and Metric measurements.
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Tutorial contains basic concepts, interactive examples, and problems with randomly generated parameters. A customer is allowed to select chapters for a test, get the test review, and save the results. Especially useful in preparing for tests.
Tutorial contains basic concepts, interactive examples, and problems with randomly generated parameters. A customer is allowed to select chapters for a test, get the test review, and save the results. Especially useful in preparing for exams. |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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[50] Homework 8: Counting [10] How many ways are there to seat 18 people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table? Justify your answer. [10] How many ordered p
Module 1: Basic Logic Theme 1: PropositionsEnglish sentences are either true or false or neither. Consider the following sentences: 1. Warsaw is the capital of Poland. 2. .3. How are you? The first sentence is true, the second is false, while th
Module 3: Proof TechniquesTheme 1: Rule of InferenceLet us consider the following example. Example 1: Read the following "obvious" statements: All Greeks are philosophers. Socrates is a Greek. Therefore, Socrates is a philosopher. This conclusion s
Module 4: Mathematical InductionTheme 1: Principle of Mathematical InductionMathematical induction is used to prove statements about natural numbers. As students may remember, we can write such a statement as a predicate set of natural numbers wher
Module 5: Basic Number TheoryTheme 1: DivisionGiven two integers, sayand , the quotientmay or may not be an integer (e.g., but ). Number theory concerns the former case, and discovers criteria upon which one candecide about divisibility
Module 8: Trees and GraphsTheme 1: Basic Properties of TreesA (rooted) tree is a finite set of nodes such that there is a specially designated node called the root. the remaining nodes are partitioned into sets is a tree. The sets disjoint s
Appendix 1Lab Exercises For A Computer Architecture CourseA1.1 IntroductionThis Appendix presents a set of lab exercises for an undergraduate computer architecture course. The labs are designed for students whose primary educational goal is learn
Appendix 1Lab Exercises For A Computer Architecture CourseA1.1 IntroductionThis Appendix presents a set of lab exercises for an undergraduate computer architecture course. The labs are designed for students whose primary educational goal is learn |
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Product Description:
written." — School Science and Mathematics. "A very fine book." — Mathematics Teacher |
Focuses on reviewing the basics of algebra, the language of mathematics, with an emphasis on working word problems. The course will include basic review of arithmetic skills: fractions, decimals, percent, and absolute value. Also included will be basic topics of algebra: first degree equations and inequalities including graphing, formulas, and problem solving, exponents and polynomials. Credit for enrollment but not toward graduation; satisfies no college requirement.
Highlights the utility of mathematics in everyday life; increasing proficiency in advanced formula and problem solving, including real world applications. Prerequisite: MTH090 or satisfactory score on the Math placement test.
Highlights the utility of mathematics in everyday life; increasing proficiency in advanced formula and problem solving, including real world applications. Prerequisite: MTH090 or satisfactory score on the Math placement test.
A course which focuses on functions (polynomial, rational, exponential, trigonometric) and their properties. There will be an emphasis on incorporating world problems. Prerequisite: MTH090 or satisfactory score on Math placement test. |
Mastering the Fundamentals of Mathematics & High School Level—Algebra I (Set)
COURSE DESCRIPTION
Mastering the Fundamentals of Mathematics & High School Level—Algebra I (Set)
Course
1
of
2:
Algebra I Professor
Professor James A. Sellers,
The Pennsylvania State University Ph.D., The Pennsylvania State University Algebra I is an entirely new course designed to meet the concerns of both students and their parents. These 36 accessible lectures make the concepts of first-year algebra—including variables, order of operations, and functions—easy to grasp. For anyone wanting to learn algebra from the beginning, or for anyone needing a thorough review, Professor James A. Sellers will prove to be an inspirational and ideal tutor. Open yourself up to the world of opportunity that algebra offers by making the best possible start on mastering this all-important subject.
Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
An Introduction to the Course
Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations.
19.
Factoring Trinomials
Begin to find solutions for quadratic equations, starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial (a binomial expression consists of two terms, a trinomial of three). Professor Sellers explains the tricks of factoring such expressions, which is a process almost like solving a mystery.
2.
Order of Operations
The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now.
20.
Quadratic Equations—Factoring
In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly. Focus on two such forms: perfect square trinomials and differences of two squares. Learning to recognize these cases makes factoring easy.
3.
Percents, Decimals, and Fractions
Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms.
21.
Quadratic Equations—The Quadratic Formula
For those cases that defy simple factoring, the quadratic formula provides a powerful technique for solving quadratic equations. Discover that this formidable-looking expression is not as difficult as it appears and is well worth committing to memory. Also learn how to determine if a quadratic equation has no solutions.
4.
Variables and Algebraic Expressions
Advance to the next level of problem solving by using variables as the building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables, and operation symbols. Also learn some tricks for translating the language of problems (phrases in English) into the language of math (algebraic expressions).
22.
Quadratic Equations—Completing the Square
After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched.
5.
Operations and Expressions
Discover that by following basic rules on how to treat coefficients and exponents, you can reduce very complicated algebraic expressions to much simpler ones. You start by using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy.
23.
Representations of Quadratic Functions
Drawing on your experience solving quadratic functions, analyze the parabolic shapes produced by such functions when represented on a graph. Use your algebraic skills to determine the parabola's vertex, its x and y intercepts, and whether it opens in an upward "cup" or downward in a "cap."
6.
Principles of Graphing in 2 Dimensions
Using graph paper and pencil, begin your exploration of the coordinate plane, also known as the Cartesian plane. Learn how to plot points in the four quadrants of the plane, how to choose a scale for labeling the x and y axes, and how to graph a linear equation.
24.
Quadratic Equations in the Real World
Quadratic functions often arise in real-world settings. Explore a number of problems, including calculating the maximum height of a rocket and determining how long an object dropped from a tree takes to reach the ground. Learn that in finding a solution, graphing can often help.
7.
Solving Linear Equations, Part 1
In this lesson, work through simple one- and two-step linear equations, learning how to isolate the variable by different operations. Professor Sellers also presents a word problem involving a two-step equation and gives tips for how to solve it.
25.
The Pythagorean Theorem
Because it involves terms raised to the second power, the famous Pythagorean theorem, a2 + b2 = c2, is actually a quadratic equation. Discover how techniques you have previously learned for analyzing quadratic functions can be used for solving problems involving right triangles.
8.
Solving Linear Equations, Part 2
Investigating more complicated examples of linear equations, learn that linear equations fall into three categories. First, the equation might have exactly one solution. Second, it might have no solutions at all. Third, it might be an identity, which means every number is a solution.
26.
Polynomials of Higher Degree
Most of the expressions you've studied in the course so far have been polynomials. Learn what characterizes a polynomial and how to recognize polynomials in both algebraic functions and in graphical form. Professor Sellers defines several terms, including the degree of an equation, the leading coefficient, and the domain.
9.
Slope of a Line
Explore the concept of slope, which for a given straight line is its rate of change, defined as the rise over run. Learn the formula for calculating slope with coordinates only, and what it means to have a positive, negative, and undefined slope.
27.
Operations and Polynomials
Much of what you've learned about linear and quadratic expressions applies to adding, subtracting, multiplying, and dividing polynomials. Discover how the FOIL operation can be extended to multiplying large polynomials, and a version of long division works for dividing one polynomial by another.
10.
Graphing Linear Equations, Part 1
Use what you've learned about slope to graph linear equations in the slope-intercept form, y = mx + b, where m is the slope, and b is the y intercept. Experiment with examples in which you calculate the equation from a graph and from a table of pairs of points.
28.
Rational Expressions, Part 1
When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials. These functions play an important role in algebra. Learn how to add and subtract rational functions by first finding their common divisor.
11.
Graphing Linear Equations, Part 2
A more versatile approach to writing the equation of a line is the point-slope form, in which only two points are required, and neither needs to intercept the y axis. Work through several examples and become comfortable determining the equation using the line and the line using the equation
29.
Rational Expressions, Part 2
Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions.
12.
Parallel and Perpendicular Lines
Apply what you've discovered about equations of lines to two very special types of lines: parallel and perpendicular. Learn how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves. Also try your hand at word problems that feature both types of lines.
30.
Graphing Rational Functions, Part 1
Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves.
13.
Solving Word Problems with Linear Equations
Linear equations reflect the behavior of real-life phenomena. Practice evaluating tables of numbers to determine if they can be represented as linear equations. Conclude with an example about the yearly growth of a tree. Does it increase in size at a linear rate?
31.
Graphing Rational Functions, Part 2
Sketch the graphs of several rational functions by first calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function. In the final exercise, you must simplify the expression in order to extract the needed information.
14.
Linear Equations for Real-World Data
Investigating more real-world applications of linear equations, derive the formula for converting degrees Celsius to Fahrenheit; determine the boiling point of water in Denver, Colorado; and calculate the speed of a rising balloon and the time for an elevator to descend to the ground floor.
32.
Radical Expressions
Anytime you see a root symbol—for example, the symbol for a square root—then you're dealing with what mathematicians call a radical. Learn how to simplify radical expressions and perform operations on them, such as multiplication, division, addition, and subtraction, as well as combinations of these operations.
15.
Systems of Linear Equations, Part 1
When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions.
33.
Solving Radical Equations
Discover how to solve equations that contain radical expressions. A key step is isolating the radical term and then squaring both sides. As always, it's important to check the solution by plugging it into the equation to see if it makes sense. This is especially true with radical equations, which can sometimes yield extraneous, or invalid, solutions.
16.
Systems of Linear Equations, Part 2
Expand your tools for solving systems of linear equations by exploring the method of solving by elimination. This technique allows you to eliminate one variable by performing addition, subtraction, or multiplication on both sides of an equation, allowing a straightforward solution for the remaining variable.
34.
Graphing Radical Functions
In previous lessons, you moved from linear, quadratic, and rational functions to the graphs that display them. Now do the same with radical functions. For these, it's important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol.
17.
Linear Inequalities
Shift gears to consider linear inequalities, which are mathematical expressions featuring a less than sign or a greater than sign instead of an equal sign. Discover that these kinds of problems have some very interesting twists, and they come up frequently in business applications.
35.
Sequences and Pattern Recognition, Part 1
Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence
18.
An Introduction to Quadratic Polynomials
Transition to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic. Learn how to use the FOIL method (first, outer, inner, last) to multiply linear terms to get a quadratic expression.
36.
Sequences and Pattern Recognition, Part 2
Conclude the course by examining more types of number sequences, discovering how rich and enjoyable the mathematics of pattern recognition can be. As in previous lessons, employ your reasoning skills and growing command of algebra to find order—and beauty—where once all was a confusion of numbers.
Whether you're a high-school student preparing for the challenges of higher math classes, an adult who needs a refresher in math to prepare for a new career, or someone who just wants to keep his or her mind active and sharp, there's no denying that a solid grasp of arithmetic and prealgebra is essential in today's world. In Professor James A. Sellers' engaging course, Mastering the Fundamentals of Mathematics, you learn all the key math topics you need to know. In 24 lectures packed with helpful examples, practice problems, and guided walkthroughs, you'll finally grasp the all-important fundamentals of math in a way that truly sticks.
Course Lecture Titles
24
Lectures
30
minutes/lecture
1.
Addition and Subtraction
This introductory lecture starts with Professor Sellers' overview of the general topics and themes you'll encounter throughout the course. Then, plunge into an engaging review of the addition and subtraction of whole numbers, complete with several helpful tips designed to help you approach these types of problems with more confidence.
13.
Exponents and Order of Operations
Explore a fifth fundamental mathematical operation: exponentiation. First, take a step-by-step look at the order of operations for handling longer calculations that involve multiple tasks—complete with invaluable tips to help you handle them with ease. Then, see where exponentiation fits in this larger process.
2.
Multiplication
Continue your quick review of basic mathematical operations, this time with a focus on the multiplication of whole numbers. In addition to uncovering the relationship between addition and multiplication, you'll get plenty of opportunities to strengthen your ability to multiply two 2-digit numbers, two 3-digit numbers, and more.
14.
Negative and Positive Integers
Improve your confidence in dealing with negative numbers. You'll learn to use the number line to help visualize these numbers; discover how to rewrite subtraction problems involving negative numbers as addition problems to make them easier; examine the rules involved in multiplying and dividing with them; and much more.
3.
Long Division
Turn now to the opposite of multiplication: division. Learn how to properly set up a long division problem, how to check your answers to make sure they're correct, how to handle zeroes when they appear in a problem, and what to do when a long division problem ends with a remainder.
15.
Introduction to Square Roots
In this lecture, finally make sense of square roots. Professor Sellers offers examples to help you sidestep issues many students express frustration with, shows you how to simplify radical expressions involving addition and subtraction, and reveals how to find the approximate value of a square root without using a calculator.
4.
Introduction to Fractions
Mathematics is also filled with "parts" of whole numbers, or fractions. In the first of several lectures on fractions, define key terms and focus on powerful techniques for determining if fractions are equivalent, finding out which of two fractions is larger, and reducing fractions to their lowest terms.
16.
Negative and Fractional Powers
What happens when you have to raise numbers to a fraction of a power? How about when you have to deal with negative exponents? Or negative fractional exponents? No need to worry —Professor Sellers guides you through this tricky mathematical territory, arming you with invaluable techniques for approaching these scenarios.
5.
Adding and Subtracting Fractions
Fractions with the same denominator. Fractions with different denominators. Mixed numbers. Here, learn ways to add and subtract them all (and sometimes even in the same problem) and get tips for reducing your answers to their lowest terms. Math with fractions, you'll discover, doesn't have to be intimidating—it can even be fun!
17.
Graphing in the Coordinate Plane
Grab some graph paper and learn how to graph objects in the coordinate (or xy) plane. You'll find out how to plot points, how to determine which quadrant they go in, how to sketch the graph of a line, how to determine a line's slope, and more.
6.
Multiplying Fractions
Continue having fun with fractions, this time by mastering how to multiply them and reduce your answer to its lowest term. Professor Sellers shows you how to approach and solve multiplication problems involving fractions (with both similar and different denominators), fractions and whole numbers, and fractions and mixed numbers.
18.
Geometry—Triangles and Quadrilaterals
Continue exploring the visual side of mathematics with this look at the basics of two-dimensional geometry. Among the topics you'll focus on here are the various types of triangles (including scalene and obtuse triangles) and quadrilaterals (such as rectangles and squares), as well as methods for measuring angles, area, and perimeter.
7.
Dividing Fractions
Professor Sellers walks you step-by-step through the process for speedily solving division problems involving fractions in this lecture filled with helpful practice problems. You'll also learn how to better handle calculations involving different notations, fractions, and whole numbers, and even word problems involving the division of fractions.
19.
Geometry—Polygons and Circles
Gain a greater appreciation for the interaction between arithmetic and geometry. First, learn how to recognize and approach large polygons, including hexagons and decagons. Then, explore the various concepts behind circles (such as radius, diameter, and the always intriguing pi), as well as methods for calculating their circumference, area, and perimeter.
8.
Adding and Subtracting Decimals
What's 29.42 + 84.67? Or 643 + 82.987? What about 25.7 – 10.483? Problems like these are the focus of this helpful lecture on adding and subtracting decimals. One tip for making these sorts of calculations easier: making sure your decimal points are all lined up vertically.
20.
Number Theory—Prime Numbers and Divisors
Shift gears and demystify number theory, which takes as its focus the study of the properties of whole numbers. Concepts that Professor Sellers discusses and teaches you how to engage with in this insightful lecture include divisors, prime numbers, prime factorizations, greatest common divisors, and factor trees.
9.
Multiplying and Dividing Decimals
Investigate the best ways to multiply and divide decimal numbers. You'll get insights into when and when not to ignore the decimal point in your calculations, how to check your answer to ensure that your result has the correct number of decimal places, and how to express remainders in decimals.
21.
Number Theory—Divisibility Tricks
In this second lecture on the world of number theory, take a closer look at the relationships between even and odd numbers, as well as the rules of divisibility for particular numbers. By the end, you'll be surprised that something as intimidating as number theory could be made so accessible.
10.
Fractions, Decimals, and Percents
Take a closer look at converting between percents, decimals, and fractions—an area of basic mathematics that many people have a hard time with. After learning the techniques in this lecture and using them on numerous practice problems, you'll be surprised at how easy this type of conversion is to master.
22.
Introduction to Statistics
Get a solid introduction to statistics, one of the most useful areas of mathematics. Here, you'll focus on the four basic "measurements" statisticians use when gleaning meaning from data: mean, media, mode, and range. Also, see these concepts at work in everyday scenarios in which statistics plays a key role.
11.
Percent Problems
Use the skills you developed in the last lecture to better approach and solve different kinds of percentage problems you'd most likely encounter in your everyday life. Among these everyday scenarios: calculating the tip at a restaurant and determining how much money you're saving on a store's discount.
23.
Introduction to Probability
Learn more about probability, a cousin of statistics and another mathematical field that helps us make sense of the seemingly unexplainable nature of the world. You'll consider basic questions and concepts from probability, drawing on the knowledge and skills of the fundamentals of mathematics you acquired in earlier lectures.
12.
Ratios and Proportions
How do ratios and proportions work? How can you figure out if a particular problem is merely just a ratio or proportion problem in disguise? What are some pitfalls to watch out for? And how can a better understanding of these subjects help save you money? Find out here.
24.
Introduction to Algebra
Professor Sellers reviews the importance of math in daily life and previews the next logical step in your studies: Algebra I (which involves variables). Whether you're planning to take more Great Courses in mathematics or simply looking to sharpen your mind, you'll be sent off with new levels of confidence. |
MATH 7: All the expected algebraic topics are covered in this text. Patterns, relations, and functions are presented early in the text and are reviewed and practiced throughout the year. Order of operations are applied to whole numbers, integers, rational numbers, and exponents. Students build on their understanding of variables and expressions and extend them to equations and inequalities. Students also analyze patterns and functions leading to graphing on the coordinate plane.
MATH 8: Similar to Course 2 however the development of algebraic thinking progresses from Course 1 to Course 3, building a solid foundation for students to have confidence and success in Algebra I.
ALGEBRA I: Saxon Algebra 1 covers advanced topics such as arithmetic of and evaluation of expressions involving signed numbers exponents and roots. Students learn properties of the real numbers; absolute value and equations or inequalities involving absolute value; unit conversions; solution of equations in one unknown and solution of simultaneous equations; polynomials and rational expressions; word problems requiring algebra; Pythagorean theorem; functions and functional notation; solution of quadratic equations; and much, much more.
GEOMETRY: Saxon Geometry books teach postulates and theorems and two column proofs. They also teach triangle congruence, surface area and volume, vector addition, slopes and equations of lines. With topics like these, Saxon Geometry books cover all the ground of a traditional high school geometry course, with some additional topics thrown in to connect with real life applications as well as Algebra review.
ALGEBRA II: Saxon Algebra 2 topics covered include: graphical solution to simultaneous equations; roots of quadratic equations, even including complex roots; inequalities and systems of inequalities; logarithms and antilogarithms; exponential equations; basic trigonometric functions; vectors; polar and rectangular coordinate systems, and so much more! There are also many different types of word problems requiring algebra in their solution, and real world applications in areas such as physics and chemistry are discussed.
Saxon Algebra 2 books are rather unique in that they not only cover second year algebra, but also a good deal of geometry, equaling about a semester's work of informal geometry, including proof outlines. There is also treatment of set theory and probability and statistics. |
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