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Mathematics is a compulsory subject until the end of Year 11 and is one of the few subjects taught banded, with high-expectation groups extended and accelerated. Students with needs in numeracy are also targeted with extra support.
Homework
Students will receive homework on a daily basis except on assessment days; this is checked and feedback is given. Homework has a tremendous impact on student achievement and thus gets taken very seriously.
Assessment
Year 9 and 10 students study the units from the 3 strands of the New Zealand Mathematics Curriculum over their schooling. Assessment is aligned as closely to the structure of NCEA as possible. So a unit may be assessed as "Internal" (a project style assessment assessed during the year) or "External" (assessed at the end of the year in an examination).
Senior courses are entered in a combination of Unit Standards (best grade is achieved) and Achievement Standards (can also achieve with Merit and Excellence)
Structure
Year 9
Four Bands. Students are assessed in term 1 and placed appropriately in 1 of these 4 bands.
Year 10
Four Bands. Students are placed according to their final results in Year 9.
Year 11
Four courses to cater for the different needs of all students. The content and assessed standards in these courses is carefully selected to be give each student the chance to achieve well and be challenged.
Year 12
Three different courses to cater for the needs of all students - regular holiday workshops
Year 13
Mathematics with Statistics : Two courses
Mathematics with Calculus : Two courses
More information about each course can be found in the student handouts that all students receive at the beginning of their course.
Maths Publication
Maths Uncensored is a delightful read, put together entirely by students and aimed at introducing Maths at Springs to the community. |
Maths Quest 7 Australian Curriculum Edition & Ebookplusaths Quest 7 for the Australian Curriculum is written specifically for the new Australian Mathematics curriculum and includes a comprehensive coverage of all the content strands: Number and Algebra, Geometry and Measurement and Probability and Statistics.The content strands are fully supported by Jacaranda's exciting new digital resources eBookPLUS to engage students and enhance understanding. What is eBookPLUS? This title features eBookPLUS which is provided FREE with the textbook, but is also available for purchase separately. eBookPLUS is an electronic version of the textbook and a complementary set of targeted digital resources. These flexible and engaging ICT activities are available to you online at the JacarandaPLUS website ( ). Your eBookPLUS resources include: • The entire textbook in electronic format • Interactivities designed to enhance students understanding of particular concepts • eLessons to bring the real world into the mathematics classroom • Weblinks to useful support material on the internet • ProjectsPLUS which provides all the tools you need to engage and challenge students in the completion of an ICT-based research project |
This early work on spherical trigonometry is both expensive and hard to find in its first edition. It contains a comprehensive account of the subject and includes numerous examples and exercises. This… |
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. |
Calculus: Single VariableCalculus: Single Variable exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativity and variety, continue to motivate and challenge students. The fifth edition includes even more problems and additional skill-building exercises.Foundations: The 5th edition of the text provides students with a clear understanding of the ideas of calculus as a solid foundation for subsequent courses in mathematics and other disciplines.
Rule of Four: Encourages students with a variety of learning styles to expand their knowledge by presenting ideas and concepts graphically, numerically, symbolically, and verbally.
Balanced Approach: The authors understand the important balance between concepts and skills. As instructors themselves, they know that the balance that an instructor chooses depends on the students they have: sometimes a focus on conceptual understanding is best; sometimes more drill is appropriate. The flexibility of the Fifth Edition allows instructors to tailor the course to their students.
Student Understanding: Exposition written in a way that students can actually read and more easily understand.
Flexible approach to technology
New to This Edition
Expanded Skills and Practice: The 5th edition includes a number new of skill-building and practice exercises, as well as additional problems.
Updated Data and Models: References to dates, prices, and other time-bound quantities have been updated for contemporary applied examples, problems, and projects. For example, Section 11.7 now introduces the current debate on Peak Oil production, underscoring the importance of mathematics in understanding the world's economic and social problems.
New Projects: There are new projects in Chapter 1:Which way is the Wind Blowing?; Chapter 5: The Car and the Truck; Chapter 9: Prednisone; and Chapter 10: The Shape of Planets.
More Problems: 10% more "problem"-type questions now included in the test banks and instructor's manuals.
Chapter 4 Reorganization: This chapter has been reorganized to smooth the approach to optimization.
New ConcepTests: Promote active learning in the classroom. These can be used with or without clickers, and have been shown to dramatically improve student learning. Available in a book or on theweb at
Expanded Appendices: A new Appendix D introducing vectors in the plane has been added. This can be covered at any time, but may be particularly useful in the conjunction with Section 4.8 on parametric equations.CourseSmart (eBook)
CourseSmart goes beyond traditional expectations-providing you instant, online access to the textbooks you need at an average savings of 50%. To learn more go to: coursesmart.com.
By offering an eTextbook option, it makes you more likely to buy your assigned textbook and to...
Resources for Instructors
Instructor's Manual to accompany Calculus: Single and Multivariable, 5th Edition
Instructor's Manual.
Printed Test Bank
Printed Test Bank access to electronic versions of the Instructor's Manual, the Instructor's Solutions Manual, additional projects, as well as other valuable resources. |
" Plain – English explanations and step-by-step guidance." Really? Well actually yes. This book gives a really nice and simple explanation (in human language) covering all the important aspects of this subject. So this is my shor review of this useful book.
Like all of the "For Dummies" books, this one is also written in a very smart and handy style. Even the very first page of the book contains handy and useful information – tables with equations of geometry and trig functions. Furthermore, throughout any chapter of the book, useful tips, warnings (yeah maths can be dangerous), critical concepts and rules are clearly marked with little icons, which are very useful. Funny illustrationsand tiny comics at the start of the chapters are also a nice way to make your journey through the land of maths a little easier. So, in conclusion, the layout is very nice, interesting, handy and especially useful for students, who are revising the subject.
Now a few words about the author. The author, Mark Ryan, has been teaching maths since 1989. He runs the Math Center in Winnetka, Illinois. His natural talent in maths was clearly shown, when he scored 800 (perfect score) on the maths part of the SAT. However, his real talent is explaining maths in plain English. This talent is truly obvious throughout the book, where hard to grasp subjects are explained in plain English with a help of helpful graphs and illustrations.
The book is organized neatly into a couple of parts regarding the most important subjects of calculus – differentiation and integration. In addition, there are a couple of other important subjects like limits, finding areas of various geometrical figures and revision in algebra and trig functions. However, it would be very useful if the book had more problems, with explained and detail answers. Nevertheless, it's still a great book to revise and deeply cover the subject of calculus.
All in all, I had fun reading this book and revising calculus, which is a major part of maths in both late high school and college. Furthermore, the book helped me to understand this subject more deeply. Thus it's a great book with a couple of minor drawbacks. So if you're looking for a great book to revise or learn calculus, look no more.
Score: 8/10
Similar books: "Physics for Dummies", "Chemistry for Dummies", "Maths for Dummies". |
Bedford3540761802. Vector calculus is the cornerstone of a vast amount of applied mathematics This is an undergraduate text. This is both concise and comprehensive in structure and f [more]
3540761802. Vector calculus is the cornerstone of a vast amount of applied mathematics This is an undergraduate text. This is both concise and comprehensive in structure and format; 8vo; 182 pages |
AP Calculus AB Curriculum Map
Freeport Public Schools
Time Topics Skills
Line (Use of graphing calculator is required on a regular basis)
Students will:
First Pre-Calculus • Simplify expressions containing absolute value
Quarter • Graph: a) y = l f (x) l
Limits b) y = F l x l
Derivative • Solve absolute value equations
• Graph and solve F (x) s a
• Graph and solve lF (x) l s a
• Determine whether a function is odd, even or neither
• Determine which, if any, symmetries a relation has
• Find limits of a function algebraically, graphically, and numerically
• Determine whether or not a function is continuous
• Be able to graph functions
• Find the derivative of a function using the definition
• Be able to find derivative graphically, algebraically, and numerically
• Be able to use the calculator to evaluate derivatives numerically
• Be able to do velocity problems algebraically and graphically
• Apply the chain rule
• Find the equation of a line tangent to the curve
• Find the equation of a line that best approximates the curve
Second Application of • Find the global and local extreme of a function
Quarter Derivatives • Be able to do application of Rolle's and mean value theorem
• Be able to graph derivative given, function, and graph function given derivative
Integration • Be able to determine where a function is increasing and decreasing
• Be able to determine the concavity of a function and its points of inflection
• Do related rates and problems
• Be able to do initial condition problems
• Use the right approximation method and left approximation method
• Make a connection between differential and integral calculus
• Find antiderivative of appropriate functions
• Be able to do velocity problems
AP Calc AB 09-10.doc FPS 2009-2010
Page1
AP Calculus AB Curriculum Map
Freeport Public Schools
Time Topics Skills
Line (Use of graphing calculator is required on a regular basis)
Students will:
Third Integration
Quarter • Find the approximation to an integral by using trapezoidal rite
Application of • Find area using integrals
Integration • Find volume
• Find derivative of natural log
Functions • Integrate with ⊥
• Differentiate and integrate with e
• Do exponential growth and decay problems
• Find derivatives and integrals of "a"
• Be able to differentiate inverse trig functions and do antiderivative problems with inverse trig in answer
• Know when and how to use L'Hospital's rule in a limit problem
• Be able to find the derivative of the inverse of a function without finding the inverse
Fourth Review • Be able to do Part I questions with and without a calculator
Quarter • Be able to do Part II with and without calculator
Final Exam
AP Calc AB 09-10.doc FPS 2009-2010
Page2 |
Development of Calculus ?
Javier Zhito asked
I need a essay o something about the Development of Calculus. Please include different sources and explain the development of Calculus including many mathematical details. Give historical background and examples. |
A combination of theoretical perspectives is used to create a rich description of student reasoning when facing a highly-geometric electricity and magnetism problem in an upper-division active-engagement physics classroo ...
A mathematical evidence-in a statistically significant sense-of a geometric scenario leading to criticality of the Navier-Stokes problem is presented. (C) 2012 American Institute of Physics. [ |
Algebra And Trigonometry - With 2 Cds - 4th edition
Summary: Bob Blitzers unique background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets students engaged and keeps them engaged. Presenting the full scope of the mathematics is just the first step. Blitzer draws students in with vivid applications that use math to solve real-life problems. These applications help answer the question When will I ever use this? Students stay engaged because the book helps them...show more remain focused as they study. The three-step learning systemSee It, Hear It, Try Itmakes examples easy to follow, while frequent annotations offer the support and guidance of an instructors voice. Every page is interesting and relevant, ensuring that students will actually use their textbook to achieve success! ...show less
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Your student's problem-solving and reasoning skills will be strengthened, while his or her understanding of math as a tool of commerce, the language of science, and a means for solving everyday problems will be developed. The biblical basis and revelence of math is made evident from beginning to end. |
a two-semester course of beginning and intermediate algebra. By requiring the use of the graphing calculator, this text joins a leading trend in the developmental math market. The authors' state their view on their use of technology as, "We use the graphing technology to enhance, not to replace, the students' mathematical skills." Elementary and Intermediate Algebra: Graphs and Model... MOREs covers both elementary and intermediate algebra without the repetition of instruction necessary in two separate texts. Geared toward helping students learn and apply mathematics by incorporating the hallmark Bittinger Five-Step problem-solving process, this text integrates real world applications with proven pedagogy and an accessible writing style. This text will be graphing calculator required but still has a strong balance of skill and drill algebra. |
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Abstract
The paper describes a pilot study on the use of computer algebra at upper secondary level. A symbolic calculator was introduced in a pre-examination class studying for advanced pre-university mathematics. With the theoretical framework of Realistic Mathematics Education and Developmental Research as a background, the study focused on the identification of obstacles that students encountered while using computer algebra. Five obstacles were identified that have both a technical and a mathematical character. It is the author's belief that taking these barriers seriously is important in developing useful pedagogical strategies |
I'm a college student who is having some issues really understanding Calculus II, which is a problem as A) I am a Comp Sci major, and B) I need to take Calc III as well, and an advanced math elective as well. I've not heard good things about the school provided tutors, (IE, one is not too skilled at math, and the other is really smug and poor pedagogically) and while I could get a private tutor to help me, I was wondering if there are any good resources for online math tutors, like there are for online language tutoring over Skype. Does a directory of such tutors exist, has anyone had a good experience doing this (maybe even with names?), and how much can I expect to pay? [more inside]
posted by mccarty.tim
on Jan 14, 2013 -
11 answers
My math skills are no where near where I'd like them to be. Can you recommend a self-paced math learning site? Pretty much any branch of math can apply. [more inside]
posted by Ookseer
on Aug 13, 2012 -
7 answers
I'm teaching myself to code (Python, mainly), and it's going well, except I keep on running into one big problem: I haven't done any math since high school (6 years ago). I used to be top of my class, and now I can't do simple word problems anymore. This gets tough because they inevitably pop up in the books I'm using, and then I stop dead in my tracks. Could anyone recommend a couple of books or web resources to help me get back up to speed? I never took Calculus, so a good Calculus book would help, too. [more inside]
posted by flibbertigibbet
on Dec 10, 2011 -
22 answers
Calculusfilter. A man is led to the center of a valuable field which he does not own. He coats his feet in blue paint so that his path can be traced. At dawn he begins walking. At a randomly selected time he will be told to stop walking, whereupon he will walk in a perfectly straight line back to the starting point. Then he will be given all the land that has been circumscribed by his blue path. [more inside]
posted by foursentences
on Aug 30, 2011 -
89 answers
The area between f(x), the x-axis and the lines x=a and x=b is revolved around the x-axis. The volume of this solid of revolution is b^3-a*b^2
for any a,b. What is f(x)? [more inside]
posted by stuart_s
on Sep 29, 2010 -
27 answers
I'm taking a calc-based physics class as well as a calculus class. The last time I took a math class was 5 years ago and I would love some refresher resources - or even better, an intensive algebra-trig-precalc course. Are there any online or in the DC area? [more inside]
posted by alaijmw
on Sep 2, 2010 -
10 answers
I need help understanding the "principle domain" (a term only my professor seems to use) of a polar function. That is, how do I find the smallest value delta such that [0,delta] plots all of the unique points on the curve; any values greater than delta re-trace points. I suspect that if MeFi could identify the more common name for this, uh, procedure, I'd be set.
posted by phrontist
on Aug 4, 2010 -
10I'm looking for an example of an alternating series: the terms of which are (-1)^n b_n, where b_n -> 0 as n -> infinity, but the sequence {b_n} is not decreasing, and the sum from n=1 to infinity diverges. [more inside]
posted by evinrude
on Apr 15, 2009 -
25 answers
Planning on teaching myself Calculus I and II in order to take the AP Calculus BC exam this May. If you've taught or taken either class, at a high school, university, or independently, read on. [more inside]
posted by Precision
on Aug 16, 2008 -
28Where can I take (or simulate taking) a basic calculus course for free or cheap. I don't care about credits or anything, I just need to defeat calculus. This time, it's personal. [more inside]
posted by dumbledore69
on Oct 25, 2007 -
14 answers
I remember reading an anecdote about Feynman, written by Feynman either in Surely, You're Joking or What Do You Care What Other People Think. He was describing how in his undergraduate years people were really impressed by his ability to do integrals, all because he knew this integration technique that wasn't taught very often. What was that technique?
posted by phrontist
on Dec 4, 2006 -
10 answers
Could someone suggest book(s) that deal with tips/tricks/methods of performing mathematical calulations faster. I mean higher mathematics and not just elementary operations like addition/multiplication/finding the root etc.
What I am looking for are hints and shortcuts that would help me with stuff like calculus, linear albegra, vectors and such.
Thanks!
posted by sk381
on Sep 20, 2006 -
7 answers
"Math 51H provides a rigorous, proof-based introduction to linear algebra and differential calculus in several variables." Recommend a book to catch me up to the starting point for this course! [more inside]
posted by devilsbrigade
on Jul 8, 2006 -
20 answers
I'd like to learn Math. I'm particularly interested in learning trig and calculus. I'm don't need to learn these disciplines for any purpose. I'm just interested. I'm a reasonably bright guy, with a logical mind (I've worked as a programmer), and I'm a good self-learner. I'm not in a rush (don't mind working at this for a few years). What books/resources would you recommend? I should probably go all the way back to Algebra, which is pretty much where I left off in High School years ago.
posted by grumblebee
on Dec 27, 2003 -
12 answers |
LearnNext CBSE Class XII Maths NCERT Solved Exercises CD's excellent navigation enables the students to easily find what they are looking forward and they move a step further to a new level of up to date awareness when calculating, analyzing, problem solving, and evaluating.
In this CD Exercises and problems are solved with explanation in a coherent and much interesting manner by expert teachers, which help you to understand the problems in a fraction of seconds. Choosing of this package results to score more in the exams and as students learn how to apply and use higher order thinking skills, students learn how to question the accuracy of their solutions and findings.
This Pack is provided with real-life explanation for all Exercises prescribed in the NCERT Maths textbooks. Here each Maths problem is solved in and demonstrated in an interesting way. Students can learn all the solutions in audio-visual form which makes the study interesting. Our CBSE Class XII Maths NCERT Solved Exercises is just the way they are explained in a real-time class room. Students can literally take part in the step-by-step solution process of all the exercises. |
infocobuild
EE 261 - The Fourier Transform and its Applications
Stanford Univ. - EE 261 - The Fourier Transform and its Applications. This consists of 30 lectures given
by Professor Brad Osgood. The Fourier transform is a tool for solving physical problems. In this course
the emphasis is on relating the theoretical principles to solving practical engineering and science problems.
Topics include: The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform
of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms.
Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis
of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and
use in imaging. Further applications to optics, crystallography. (from see.stanford.edu)
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EE 261 - The Fourier Transform and its Applications Instructors: Professor Brad Osgood. Handouts. Assignments. Exams. The Fourier transform is a tool for solving physical problems. In this course the emphasis is on relating the theoretical principles to solving practical engineering and science problems. see.stanford.edu/see/courseinfo.aspx?coll=84d174c2-d74f-493d-92ae-c3f45c0ee091 |
a good knowledge of Linear Algebra, especially the theory of orthogonal transformations in real Euclidean
spaces;
the following basic notions of Group Theory: groups, the order of a
finite group, subgroups, normal subgroups and factor (or quotient) groups,
homomorphisms and isomorphisms, permutations (standard notation for
them and rules for their multiplication), cyclic groups, action of a
group on a set.
Specification
Aims
This lecture course aims to introduce students to the classification and construction of finite
reflection groups.
Brief Description of the unit
Finite reflection groups constitute one of the most important and
frequently used classes of groups. The development of the theory in the
course will be purely geometrical; the necessary facts from
geometry (hyperplane arrangements, convex cones and polytopes) will be
introduced as the need arises.
Learning Outcomes
On successful completion of this course unit students will
know the classification of regular polytops in three dimensions and their symmetry groups; |
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Mathematics Courses
MAT 003 - Intermediate Algebra
3 sh
The topics include a review of the real number system, linear equations and applications, linear inequalities and absolute value, graphs and functions, exponents, polynomial functions, factoring, rational functions, root functions, exponential and logarithmic functions. Students who have previously received credit for a higher-numbered mathematics course may not receive credit for this course without permission of the instructor.
MAT 005 - Mathematics for Nurses
1 sh
This course is an introduction to computing techniques used by nurses in hospitals and other clinical situations. Topics include a review of arithmetic and basic computational techniques, standard and apothecary measurement systems, dimensional analysis, and one-factor and multi-factor medication problems. These topics will be examined in depth with a focus placed on understanding the underlying computational techniques. Prerequisite: None.
MAT 017 - Introduction to Mathematics
3 sh
This General Education introductory-level course is intended to acquaint the student with the nature and spirit of mathematics. Topics include set theory, logic, counting methods, probability, statistics, and algebra-based problem-solving with graphical and analytic solutions.
MAT 030 - Survey of Mathematics
3 sh
Sets and logic; number systems; relations and functions; introduction to matrices; linear systems; counting and probability; sequences and limits; introduction to differential and integral calculus.
MAT 040 - Geometry
3 sh
An informal, intuitive study of topics in geometry. Non-metric geometry of the plane and space; measurement; error in measuring; simple closed curves; area; congruence; similarity; graphing in the plane and space; modern geometries; groups of geometric transformations. Open to all majors.
MAT 045 - Women in Mathematics
3 sh
This course examines women who have made significant contributions to the field of mathematics. Both their lives and their work will be explored as well as the gender issues surrounding their endeavors. Furthermore, mathematical topics related to their contributions will be discussed. Prerequisite: 2 years of high school algebra.
MAT 103 - Fundamentals of Math I
3 sh
This is the first course in a two-course sequence that is required for all Elementary Education and Special Education majors. It is not open to other majors. Topics include problem-solving; logic; set theory; mathematical systems; systems of numeration; number theory; equations and inequalities; and properties of whole numbers, integers, rational numbers, and real numbers. MAT 103 is a prerequisite for MAT 104 and ELU 308.
MAT 104 - Fundamentals of Math II
3 sh
This is the second course in a two-course sequence that is required for all Elementary Education and Special Education majors. It is not open to other majors. Topics include informal geometry; measurement; probability; statistics; and computer applications. MAT 103 is a prerequisite for MAT 104. MAT 103 and MAT 104 are prerequisites for ELU 308.
MAT 105 - College Algebra
3 sh
This course is intended for students with an elementary knowledge of algebra who need more work in algebraic topics before taking more advanced mathematics courses. Topics include properties of the real numbers, problem-solving using equations and inequalities, algebraic functions, graphing, and systems of equations. A graphing calculator is required for this course. Prerequisite: Two years of algebra at the high school level.
MAT 106 - Trigonometry
3 sh
This course is intended for students with an elementary knowledge of algebra who need more work in trigonometric topics before taking more advanced mathematics courses. Topics include properties of and operations with functions, inverse functions, exponential and logarithmic functions, angle measurement, trigonometric functions and their inverses, graphing functions, and problem-solving with equations that use the functions covered in the course. A graphing calculator is required for this course. Prerequisite: MAT 105 or its equivalent.
MAT 115 - Precalculus Mathematics
3 sh
This course is designed to give students a thorough review of the mathematics background needed for calculus courses. The course covers all the topics listed in the descriptions of MAT 105 and MAT 106. A graphing calculator is required for this course. Prerequisite: Three years of high school mathematics in algebra and trigonometry.
MAT 121 - Mathematics for Business and Information Science
3 sh
This course focuses on the application of mathematical concepts and methods to problems that arise for students who major in Business or Computer Science. The topics include a review of algebraic concepts and problem solving, systems of linear equations, matrix algebra, linear programming with graphical and simplex method solutions, and probability.A graphing calculator is required for this course. Prerequisite: MAT 105 or two years of high school algebra.
MAT 122 - Calculus for Business and Information Science
3 sh
This course focuses on the application of concepts and methods of calculus to problems that arise for students who major in Business or Computer Science. The topics include differential and integral calculus; applying derivatives, differentials, and integrals to problem-solving; and maximizing and minimizing functions. A graphing calculator is required for this course. Prerequisite: MAT 121.
MAT 140 - Applied Statistical Methods
3 sh
An introduction to quantitative methods in the behavioral sciences. Techniques for obtaining, analyzing and presenting data in numerical form; measures of central tendency and dispersion; normal distribution curve; standard scores; applicability of probability and sampling theory to research in the behavioral sciences; interpretation of confidence intervals; hypothesis testing; correlation; linear regression. A graphing calculator is required for this course. Prerequisite: MAT 105 or two years of high school algebra.
MAT 171 - Calculus I
3 sh
This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include the definition and calculation of limits, continuity and differentiability, differentials, derivatives of algebraic and trigonometric functions, the application of derivatives to graphing, antiderivatives, and the introduction of the definite integral. A graphing calculator is required for this course. Prerequisite: Three years of mathematics in algebra and trigonometry.
MAT 172 - Calculus II
3 sh
This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include the definition, properties, and applications of definite integrals, properties, derivatives, and integrals of exponential, logarithmic, trigonometric, inverse trigonometric, and hyperbolic functions with applications; and techniques of integration. A graphing calculator is required for this course. Prequisite: Completion of MAT 171 with a grade of C or higher.
MAT 175 - Mathematical Investigations
3 sh
This course is applicable in the Honors Program and is open to any student in the Honors Program as well as to any student having completed at least 15 s.h. of study at Kutztown University with a grade point average of 3.00 or greater. This course, divided into three to five segments, providesfor study of concepts, procedures and applications in several mathematical disciplines. Topics studied will be drawn from the following areas: mathematical thought processes, history of mathematics, mathematics of finance, statistics, operations research, number theory, graphs as mathematical models, and finite geometries. This course cannot be taken for credit by mathematics majors in Secondary Education or the College of Liberal Arts and Sciences. Credit for the course can be applied in Categories IV or V in General Education. PREREQUISITES: at least three years of high school academic mathematics, including trigonometry, or permission of the department chair.
MAT 220 - History of Mathematics
3 sh
A study of mathematics as it has developed through the centuries and the mathematicians who have contributed to its growth. Mathematics of early Babylonian and Egyptian civilizations; mathematics under Greek influence; Chinese, Hindu and Arabic contributions; the Renaissance period; Seventeenth and Eighteenth Century mathematics; the liberation of geometry and arithmetization of analysis of the Nineteenth Century; Twentieth Century mathematics. Not applicable toward the Arts and Sciences Mathematics major. Prerequisite: Completion of MAT 171 with a grade of C or higher.
MAT 224 - Foundations of Higher Mathematics
3 sh
This course is designed to prepare the student for the study of advanced mathematics. Topics include fundamentals of logic, proof strategies, the algebra of sets; relations, including equivalence relations; functions and their properties; countable sets and counting techniques; ordered and well-ordered sets. This course should be taken only after the student has taken at least two college-level mathematics courses. Prerequisite: Completion of MAT 171 with a grade of C or higher.
This course is designed for students who have, in addition to an interest in geometry, some previous experience in this subject area, either on the high school or college level. Topics include Euclidean geometry using Hilbert's axioms; neutral geometry; the historical development of non-Euclidean geometries; and hyperbolic geometry. Prerequisite: Completion of MAT 224 with a grade of C or higher.
MAT 260 - Linear Algebra
3 sh
This course gives the student an opportunity to make an in-depth investigation of a specialized area of mathematics which has wide-spread practical applications in the arts and sciences but still allows work with abstract concepts. A study of the properties of vector spaces; matrix theory with applications using systems or equations and determinants; linear transformations and invariants under such mappings. Prerequisite: Completion of MAT 224 with a grade of C or higher.
MAT 273 - Calculus III
3 sh
This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include indeterminate forms and improper integrals; sequences, series, and convergence tests; differentiation and integration of power series; conic sections; polar coordinates and polar integrals; vectors in two and three dimensions; operations on vectors; limits, derivatives and integrals of vector functions. A graphing calculator is required for this course. Prerequisite: MAT 172.
MAT 274 - Calculus IV
3 sh
This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include three-dimensional surfaces; the definition, properties, and partial differentiation of functions in more than one variable with applications; finding the extrema of functions in two variables; Lagrange multipliers; multiple integrals in various coordinate systems; Jacobians; line integrals in vector fields; and the application of Green's Theorem, the Divergence Theorem, and Stokes' Theorem. A graphing calculator is required for this course. Prerequisite: MAT 273.
MAT 280 - Cooperative Internship in Mathematics
3-6 sh
The internship consists of 6 to 12 weeks of full-time employment that provides students with a supervised industrial experience in mathematics. The internship is supervised by a member of the Mathematics Department. This internship is available only to Mathematics majors, and is taken on a pass/fail basis. PREREQUISITE: Substantial completion of the required and concomitant courses in the Mathematics major with an above-average grade-point average. Approval by the department chair is required. 3 to 6 credits
Numerical methods fundamental to scientific computing are developed. Topics include finite difference calculus; zeros of a function; matrix computations; solutions to systems of linear equations; approximation by polynomials; numerical differentiation and integration; numerical solutions of ordinary differential equations; rounding errors and other types of errors. Selected algorithms will be run on the computer. Prerequisite: MAT 260 and MAT 273.
MAT 340 - Differential Equations
3 sh
Theory and methods of solving ordinary differential equations are investigated - equations include first order, linear and systems; methods of solutions include exact, substitution, reduction, undetermined coefficients and variation of parameters. Consideration is given to application to the physical and natural sciences. Prerequisite: MAT 273.
MAT 351 - Advanced Calculus I
3 sh
Introduction to the structure of the real number system and its topology; metric space and its topology; basic theorems of real analysis; differentiable functions. Prerequisite: Completion of MAT 224 and 273 with a grade of C or higher.
MAT 352 - Advanced Calculus II
3 sh
Introduction to the theory of Reimann-Stieltjes integration; functions of bounded variation; Lebesgue measure and Lebesgue integrals; uniform convergence of sequences and series of functions. Prerequisite: MAT 351.
MAT 361 - Mathematical Methods in Operations Research
3 sh
Operations Research uses quantitative methods to determine the best decision for an operating system. A mathematical approach to studying methods as applied to the decision process in industry is taken. The methods studied are selected from among linear programming; game theory; mathematical programming; graph theory and network analysis; and queuing theory. Prerequisite: MAT 121 or MAT 260 or permission of the instructor.
MAT 370 - Selected Topics in Mathematics
3 sh
This course involves individual or small group independent study in some area of mathematics under the direction of a mathematics staff member. This study can be made in any area of mathematics or mathematical application. A student may register for this course more than once up to a maximum of six semester hours of credit. Prerequisite: The consent of the student's advisor, instructor and department head and their approval of the project.
MAT 372 - Independent Study and/or Projects in Mathematics
6 sh
This course involves individual or small group independent study in mathematics under the direction of a mathematics staff member. This study can be carried out in any area of mathematics or its application. A student may register for this course more than once up to a maximum of six semester hours of credit. Prerequisites: approval of the student's advisor, the instructor, and the department chairperson.
MAT 380 - Senior Seminar in Mathematics
3 sh
Readings and discussions in areas of student interest and background. The student reviews and structures the mathematics he/she has learned and also explores mathematical topics not covered in the usual course offerings. The comprehensive examination for Arts and Sciences Mathematics majors is given in conjunction with this course. Required of all arts and sciences mathematics majors.
MAT 431 - Topology I
3 sh
Topics covered for this course include set theory; functions; metric spaces; basic topological concepts; topologies and neighborhood systems; open and closed sets; accumulation points and closures; bases and subbases for a topology; separation and connectedness; nets; continuity and homeo-morphisms; compactness; product and quotient spaces. Prerequisite: MAT 272 and MAT 311 or their equivalents.
MAT 512 - Foundations of Mathematics
This course is intended to broaden and deepen the beginning graduate student's knowledge of the foundational concept of mathematics. Topics covered are: mathematical logic, theory of sets, algebra of sets, relations and functions, ordering, equivalence classes, real numbers, and ordinal and cardinal numbers and related topics. The implementation of proof strategies and procedures are emphasized. Required of all M.Ed. mathematics majors.
MAT 540 - Theory of Probability
The topics in this course include axiomatic probability, probability spaces, conditional probability, random variables and functions of random variables, probability distributions, sums of random variables, and the Central Limit Theorem are studied. Prerequisites: MAT512 or its equivalent.
MAT 545 - Statistical Inference and Sampling Theory
The topics in this course include the Weak Lay of Large Numbers, estimation of parameters, Central Limit Theorem, confidence intervals, regression analysis, sampling from a normal population, and testing hypotheses. Prerequisites: MAT540 or its equivalent.
MAT 550 - Foundations of Geometry
The topics in this course include foundational aspects of geometry, postulational systems and their properties, Euclidean geometry from both the metric and the synthetic viewpoints, finite geometries, non-Euclidean geometries, and geometric transformations are studied. Prerequisites: MAT512 or its equivalent.
The topics in this course include rings, homomorphisms, ideals and quotient rings, field of quotients of an integral domain, Euclidean rings, polynomial rings, vector spaces, extension fields, constructability criteria, Galois Theory, solvability of radicals. Prerequisites: MAT561 or its equivalent.
MAT 580 - Special Topics in Mathematics
This course is designed to enable the student to pursue interests in some area of mathematics. The function of this course is not to introduce the student to beginning concepts. Rather, it is to permit the student to pursue the study of topics encountered in courses already taken. Prerequisites: At least two graduate courses in Mathematics. Permission of the instructor, advisor, and department chairperson. |
Complex Analysis: Theodore Gamelin's Complex Analysis.Probably the single most user friendly text on the subject there is. Wonderfully written,TONS of examples and covers an enormous breadth of topics.There are lots of good ones on this topic,but for self study,there's probably none better then this one. My one complaint is that Gamelin is sometimes TOO gentle where a proof instead of a picture would be more appropriate. But then the book is designed to be read by a vast audience from freshman to PHD level,so he can be forgiven. |
COURSE DESCRIPTION This course is designed for the student who has completed Algebra and Geometry but would benefit from a review of topics before taking Advanced Algebra with Trigonometry. Topics covered include the major concepts of both Algebra and Geometry. This is not a core mathematics class. A scientific calculator is required. |
Questions About This Book?
The Used copy of this book is not guaranteed to inclue any supplemental materials. Typically, only the book itself is included.
Summary
Building on the success of its first three editions, the Fourth Edition of this market-leading text covers the important principles and real-world applications of plane geometry, with additional chapters on solid geometry, analytic geometry, and an introduction to trigonometry. Strongly influenced by both NCTM and AMATYC standards, the text takes an inductive approach that includes integrated activities and tools to promote hands-on application and discovery. New! Tables provide visual connections between figures and concepts and help students better assess their level of mastery and test readiness. New! Chapter Tests have been added to the end of every chapter. New! Proofs have been varied to include written and visual proofs, as well as comparisons, to support students with different learning styles. New! Exercise sets in the Student Study Guide, with cross-references to the text, offer additional practice and review. New! Technology-related margin features encourage the use of the Geometer's Sketchpad, graphing calculators, and further explorations. New! Coverage now includes Section 2.6, Symmetry and Transformations. New! Technology Package includes mathSpace tutorial CD and the HM ClassPrep CD with computerized test bank. Updated! The number of Exercises and Explorations has been increased. Highly visual approach begins with the presentation of an idea, followed by the examination and development of a theory, verification of the theory through deduction, and finally, application of the principles to the real world. Discovery features reinforce the text's inductive approach: activities integrated throughout enable students to discover geometry concepts on their own, and section tools provide with hands-on application of geometric concepts Applications reinforce the connection of geometry to the real world: high-interest Chapter Openers introduce the principal notion of the chapter and relate to the real world and A Perspective On... sections conclude each chapter, providing sketches that are interesting, sometimes historical, and always informative. Summaries of constructions, postulates, and theorems are provided, and an easy-to-navigate numbering system for postulates and theorems provides a user-friendly structure. In response to user feedback, paragraph proofs feature more prominently in this edition. Comprehensive appendices include Algebra Review and An Introduction to Logic. A glossary of terms, a summary of applications in the text, and selected answers are also provided in the back of the text. |
MATHEMATICS - Students must complete 3 years of high school math to fulfill their graduation requirements. The State Education Department requires students to pass a Mathematics Regents Exam before they can graduate.
Algebra 1 - This is a one year course leading to the Integrated Algebra Regents Exam in June. The focus is on the algebra content strand. Students will explore linear equations, quadratic functions with integral coefficients, absolute value, and exponential functions. Coordinate Geometry will be integrated as well as some Trigonometry. Student must earn an average of 80% or better in the previous year of Math to enter this class. Teacher recommendation and assessment results may also be considered in course placement. The Regents Exam is the final exam for the course. (1 year, 1 credit)
Algebra 1A - Integrated Algebra 1A is the first year of a two year program. The focus of this course is on the algebra content strand. During the two years, students will develop the skills and processes needed to successfully solve problems in a variety of settings. Students will explore topics in linear equations, quadratic functions with integral coefficients, absolute value and exponential functions. Coordinate geometry will also be integrated throughout this course. Students will be required to work between US Customary and metric systems and conduct data analysis. Right triangle trigonometry will be introduced during this course. Placement in this course will be based on previous school year recommendation and assessments as applicable. Students will take a local final exam at the conclusion of the course. Successful students will take Algebra 1B the following school year and challenge the Algebra I Regents Exam in June of the second year. (1 year, 1 credit)
Algebra 1B - This is the second year of a two year program leading to the Integrated Algebra Regents Exam in June. Students will explore topics from the Algebra curriculum not addressed in the prerequisite Algebra 1A course. Placement in this course is based on successful completion of Algebra 1A or teacher recommendation. Students will take a local final exam at the conclusion of the course.(1 year, 1 credit)
Geometry - This is a one year course leading to the Integrated Geometry Regents Exam in June. The focus is on the Geometry content strand. Topics will include Quadratic Equations, Coordinate Geometry, Logic, Deductive Proofs, and Quadrilaterals. Students must earn an average of 80% or better in Algebra and a score of 80% or better on the Integrated Algebra Regents Exam. A teacher recommendation may also be considered in course placement. This course will have a local final exam. (1 year, 1 credit)
Geometry A - This is the first year of a year-and-half sequence leading to the Integrated Geometry Regents Exam in January of the second year. The focus is on the geometry content strand. Topics will include Quadratic Equations, Coordinate Geometry, and Quadrilaterals. This course will have a local final exam. Upon successful completion of this course, students may move on to Geometry B with Algebra IIA. (1 year, 1 credit)
Geometry B with Algebra II A - This is the second year of a year-and-a-half program leading to the Geometry Regents. The first half of the course will be the final third of the Geometry curriculum with the students taking the Regents in January. The second half of the course will be the first third of Integrated Algebra II with Trigonometry curriculum. This course will have a local final exam. Upon successful completion of this course, students may move on to Algebra IIB with Trigonometry. (1 year, 1 credit)
Algebra 2 and Trigonometry - This is a one year course leading to the Algebra 2 and Trigonometry Regents exam in June. This course is designed for students who have successfully completed the Geometry course. The focus is on Algebra/Trigonometry strand. Topics include Algebra, Complex Numbers, Exponential Functions, Logarithmic Functions, Rational Functions, Probability, Statistics, Sequences and Series. Students must earn an average of 80% or better in Geometry and a score of 80% or better on the Geometry Regents Exam. A teacher recommendation may also be considered in course placement. This course will have a local final exam. (1 year, 1 credit)
Algebra 2B with Trigonometry - This is the final portion of the curriculum leading to the Integrated Algebra 2 and Trigonometry Regents, which will be taken in June. Topics will include Logarithms and Exponential Functions, Trigonometric Functions, and Probability and Statistics. This course will have a local final exam.
Applied Mathematics - This course is designed as an alternative for the required third credit in Mathematics. Algebra, Geometry and Trigonometry topics are studied in the context of real-life applications using a variety of software and technology. A combination of teacher prepared materials and textbooks will be used. Entrance into this course is based on teacher recommendation.(1 year, 1 credit)
Algebra 3/Trigonometry - This course is designed to help prepare students for calculus and other college level mathematics courses. Topics include Coordinate Geometry, Conic Sections, Polynomial Functions, Trigonometric Applications, and Applications of Exponential and Logarithmic Functions. Graphing of these functions is stressed, as well as solution of problems by the aid of a scientific calculator. As time permits, additional topics such as History of Mathematics, Statistics, Vectors, and Sequences are discussed. Student projects are encouraged.
College Pre-Calculus or College Calculus - beginning with the 2010-2011 academic year, any student wishing to enroll in either College Pre-Calculus or College Calculus must take the appropriate placement test and meet the minimum score to qualify for enrollment in these courses. Students will not be able to register for these courses unless the minimum scores, as set by CCC, have been achieved, (see below).
Students wishing to enroll in Pre-Calculus must have a math score between 86 and 102.
Students wishing to enroll in Calculus must have a math score above 102.
College Pre-Calculus - This course completes the study of algebraic and trigonometric skills necessary for the successful study of calculus. Trigonometric functions and identities are applied to analytical geometry. Applications of oblique triangle trigonometry and vectors are emphasized. Systems of equations and inequalities are solved using algebraic, graphical and matrix/determinant methods. Students who are accelerated in mathematics will be eligible to take College Pre-Calculus during their junior year. Seniors who have earned an 80 or higher in Algebra II with Trigonometry will also be eligible for this course. Teacher recommendation and assessment results may also be considered in course placement. Students can earn up to three college credits for this course. Cost for this course is $50.00 per credit hour.
Calculus - College Calculus is a fifth year course and focuses on Calculus. Students can earn up to 4 college credits for this course. The cost is $50.00 per credit hour. Successful completion of College Pre-Calculus with an 80 average or teacher recommendation is a prerequisite. Students will be permitted to enter Calculus directly from Algebra 2 and Trigonometry under the following conditions:
a. Students will have earned a grade of 90 or higher for the year.
b. Students will complete an independent study during the fourth marking period in units covered in Pre-Calculus that are critical to success in Calculus.
c. Students must receive a recommendation from their Algebra 2 and Trigonometry teacher. It should be noted that Calculus is the highest Math course offered at Weedsport High School. Students who complete Calculus prior to their senior year will not be able to take a Math course at Weedsport during their senior year. |
logarithms, inverse, square, square root, power, etc |
It gives instruction and practice in the use of the TI-83+/TI84+ family of graphing calculators for advanced algebra and trigonometry skills. It has been used by dozens of schools in New York State as a course, integrated into the 11th grade math course, or as remediation.
Students learn more than how to push buttons. This workbook takes them into the math behind how the graphing calculator works and teaches how to most effectively use this powerful tool. It also includes an introduction to programming the calculator.
Math teachers and college students have also found it very useful in maximizing their use and understanding of technology.
The book includes instructions, review, and questions taken from the NYS Math B Regents questions.
The material has been praised as easy to understand and has been credited with raising student success level on the NYS Math B Regents exam.
Graphing Calculator for NYS Math A… and Beyond was developed as the curriculum for a ½ year or 1 semester math elective course.
It gives instruction and practice in the use of the TI-83+/TI-84+ family of graphing calculators. It has been used by dozens of schools in New York State as a course, integrated into the 9th or 10th grade math course, or as remediation.
Students learn more than how to push buttons. This workbook takes them into the math behind how the graphing calculator works and teaches how to most effectively use this powerful tool.
Math teachers and college students have also found it very useful in maximizing their use and understanding of technology.
The book includes instructions, review, and questions taken from the NYS Math A Regents questions. The material has been praised as easy to understand and has been credited with raising student success level on the NYS Math A Regents exam. |
Displays lines and surfaces defined algebraically in 3D space in many forms, including z=f(x,y), cylindrical polar coordinates, and parametric definitions with one (giving a line) and two (surface) parameters. View controls move the viewpoint through 3D space, using keyboard and mouse. There are options to display a surface, a mesh or a combination. The number of 'steps' on each edge (level of detail) can be controlled.
Displays graphs of algebraic functions in a variety of forms. These include polar and cartesian co-ordinates, parametric and implicit functions. A wide range of functions are built-in, from simple trig and hyperbolic functions to things such as the ceil and gamma functions. On-screen HTML help is bulit-inGraphSight is a feature-rich comprehensive 2D math graphing utility with easy navigation, perfectly suited for use by high-school an college math students. The program is capable of plotting Cartesian, polar, table defined, as well as specialty graphs. Importantly, it features a simple data and formula input format, making it very practical for solving in-class and homework problems. The program comes with customizable Axis options, too |
Maths
To support students enrolled in maths and statistics courses, The Learning Centre provides a range of activities associated with mathematics and learning skills.
Semester 2, 2013 workshops
Success in Maths for Statistics (SIMS)
Topics include formulas, arithmetic, calculator, basic statistics, graphing. See the workshop program (PDF*68kb). Complete the online readiness testing (UConnect username and password required) or complete the first CMA on your STA2300 course webpage, to self assess your knowledge and determine whether you need to attend this workshop. Completing the first CMA is part of your first assignment in STA2300.
You will need to bring a copy of the SIMS workbook (PDF* 1.34mb) with you to the workshop. If you are unable to attend a workshop, the workbook will still be useful for your data analysis studies. |
Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument. (Contains 6 tables.)Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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Full Abstract
Abstract:
In 2004, the pattern in academic pathways for high school students in the USA showed that students were completing more demanding mathematics courses. Despite the upward pattern in advanced-level mathematics course-taking, disparities among racial/ethnic groups persisted between 1982 and 2004. Using data from the Education Longitudinal Study of 2002 (ELS: 2002; Ingels et al., 2007), the current study sought to advance understanding of gender and ethnic differences in advanced mathematics course-taking. Furthermore, this study examined how the differences are related to science, technology, engineering, and mathematics (STEM) pathways in college. Results showed that the relationships between exploratory factors (both individual- and school-level factors) and advanced mathematics course-taking and STEM choices differed across ethnicity and gender. This highlights the need for further research that disaggregates data by both ethnicity and gender. (Contains 3 tables and 2 figures.) purpose of this study was to investigate the tonal perception and restoration of thirds within power chords with the instruments and sounds idiosyncratic to the Western rock/pop genre. Four separate chord sequences were performed on electric guitar in four versions; as full chord and power chord versions as well as under both clean-tone and distortion effect versions. Undergraduate music majors (N = 50) listened to all 16 chord progressions and rated their perception of the tonality ("majorness" or "minorness") in the terminal chord for each sequence, utilizing a 7-point semantic differential scale ranging from minor (1) to major (7) with a neutral indicator located in the middle (4). Participants had completed a mean 3.82 (SD = 0.66) semesters of ear training and 28 indicated they played the guitar. A three-way repeated measures ANOVA revealed significant differences between responses for chord sequences (1-4), as well as a significant interaction between chord sequences, distortion (clean versus distortion), and type of chords in the progression (whole chords versus power chords). Further analysis of data indicated that participants tended to perceive terminal power chords as major, especially when progressions were comprised of power chords and contained distortion. (Contains 1 table and 1 figure.)Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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Abstract:
This study investigated the application of the speeded lexical decision task to L2 aural processing efficiency. One-hundred and twenty Japanese university students completed an aural word/nonword task. When the variation of lexical decision time (CV) was correlated with reaction time (RT), the results suggested that the single-word recognition task could be successfully applied to the L2 aural modality. However, when CV and RT were correlated with L2 listening ability, a statistically significant relationship was not found. It was concluded that the single-word word recognition task might not be suited to measuring L2 aural word recognition efficiency. (Contains 1 table.)Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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The purpose of this article is to investigate student perceptions of college departure in three state universities in Turkey. Since the beginning of the 1990s, higher Education System in Turkey went through a massification of higher education. The rapid growth brought enrollment and dropout issues in the system. A total of 58 participants were included in the qualitative study. Of the 58 participants, 35 students were later enrolled at various other programs. Twenty-three students did not drop out. Results indicated that student retention has been an uncharted territory in Turkey. High levels of dropouts lead to inefficiencies in the higher education system. (Contains 2 tables and 1 figure.) |
Mathematics for High School Teachers- An Advanced Perspective, CourseSmart eTextbook
Description
For algebra or geometry courses for teachers; courses in topics of mathematics; capstone courses for teachers or other students of mathematics; graduate courses for practicing teachers; or students who want a better understanding of mathematics.
Filling a wide gap in the market, this text provides current and prospective high school teachers with an advanced treatment of mathematics that will help them understand the connections between the mathematics they will be teaching and the mathematics learned in college. It presents in-depth coverage of the most important concepts in high school mathematics: real numbers, functions, congruence, similarity, and more.
Table of Contents
INTRODUCTION.
1. What is Meant by "An Advanced Perspective"?
I. ALGEBRA AND ANALYSIS WITH CONNECTIONS TO GEOMETRY.
2. Real Numbers and Complex Numbers.
3. Functions.
4. Equations.
5. Integers and Polynomials.
6. Number System Structures.
II. GEOMETRY WITH CONNECTIONS TO ALGEBRA AND ANALYSIS.
7. Congruence.
8. Distance and Similarity.
9. Trigonometry.
10. Area and Volume.
11. Axiomatics and Euclidean Geometry |
Analyze statistical studies and demonstrate knowledge of the basic concepts of probability.
Demonstrate the ability to synthesize quantitative data by putting numbers in perspective, by making reasonable estimates, by using mathematical models to solve applications and by solving right triangle applications.
Develop an understanding that mathematics is meaningful and recognize the connections between mathematics and other disciplines. |
aths In Action - Higher Mathematics
Synopsis
In this title, chapter reviews, summaries and revision exercises develop students' learning. It includes: abundant questions for practice reinforcement and consolidation, differentiated questions that ensure progression, a complete set of answers to save time, and activities throughout for using graphical calculators. |
Mathematics 3, Web-Based
Mathematics is a central subject in primary schools and in teacher education programmes. Teacher education programmes in mathematics cover several aspects of the subject. For instance, in addition to the pure knowledge of the subject, there is also a special focus on its unique characteristics and their effect on the teaching of mathematics.
Module 1: Number theory and didactics
Module 2: Geometry, vectors, linear algebra
The modules comprise 15 ECTS credits and run in parallel over two semesters, and may be taken individually.
The study programme is designed for teachers in schools.
Career opportunities/Further studies
The study programme is suitable for teachers in primary and secondary education. |
Analytic Trigonometry With Application - 9th edition
Summary: Featuring updated content, vivid applications, and integrated coverage of graphing utilities, the ninth edition of this hands-on trigonometry text guides readers step by step, from the right triangle to the unit-circle definitions of the trigonometric functions. Examples with matched problems illustrate almost every concept and encourage readers to be actively involved in the learning process. Key pedagogical elements, such as annotated examples, think boxes, cautio...show moren warnings, and reviews, help readers comprehend and retain the material |
This text presents MATLAB both as a mathematical tool and a programming language, giving a concise and easy to master introduction to its potential and power. Stressing the importance of a structured approach to problem solving, the text gives a step-by-step method for program design and algorithm development. The fundamentals of MATLAB are illustrated throughout with many examples from a wide range of familiar scientific and engineering areas, as well as from everyday life.
Features:
Includes MATLAB Version 7.2, Release 2006a
Numerous simple exercises provide hands-on learning of MATLABs functions
A new chapter on dynamical systems shows how a structured approach is used to solve more complex problems.
Common errors and pitfalls highlighted
Concise introduction to useful topics for solving problems in later engineering and science courses: vectors as arrays, arrays of characters, GUIs, advanced graphics, simulation and numerical methods
Text and graphics in four colour
Extensive instructor support
Essential MATLAB for Engineers and Scientists is an ideal textbook for a first course on MATLAB or an engineering problem solving course using MATLAB, as well as a self-learning tutorial for students and professionals expected to learn and apply MATLAB for themselves.
Additional material is available for lecturers only at This website provides lecturers with:
# A series of Powerpoint presentations to assist lecture preparation
# Extra quiz questions and problems
# Additional topic material
# M-files for the exercises and examples in the text (also available to students at the books companion site)
# Solutions to exercises
# An interview with the revising author, Daniel Valentine
Numerous simple exercises give hands-on learning
A chapter on algorithm development and program design
Common errors and pitfalls highlighted
Concise introduction to useful topics for solving problems in later engineering and science courses: vectors as arrays, arrays of characters, GUIs, advanced graphics, simulation and numerical methods
A new chapter on dynamical systems shows how a structured approach is used to solve more complex problems.
Text and graphics in four colour
Extensive teacher support on solutions manual, extra problems, multiple choice questions, PowerPoint slides
Companion website for students providing M-files used within the book |
I enjoy making math less painful for my students.In Algebra I, students explore variables, and their uses. This includes solving equations and inequalities, graphing, systems of equations, and factoring. Students will also learn about rational numbers, polynomials, roots, quadratic and exponential functions |
About The Teacher
NAME:
Sue-Ann Hershey
SCHOOL:
Paxon School for Advanced Studies
CLASS:
IB Pre-Calculus & Honors and Algebra 2
Ms. Hershey has a B. S. degree in secondary education mathematics from Ball
State University. From Ball State University's website, "Ball State Teachers
College has a long and distinctive history of preparing educators and other
professionals. Founded in 1918, it has evolved into one of the most
comprehensive and renowned colleges of education in the nation."
If you would like to read about Ball State Teacher's College ranking
CLICK HERE
Ms. Hershey also holds a Master's of Arts in Math Education from Ball State
University.
As she enters her 37th year of teaching, Ms. Hershey is even more excited
about this school year than any other year. This is going to be a great
year!!! Remember: Maximum Effort = Maximum Return
Mission
Course Description for Pre-Calculus:
Success in college level mathematics (including calculus) begins with a good
understanding of algebra and trigonometry. Although we review some of the
basic concepts developed in algebra, we assume that students in this course
have an exemplary background of two years of algebra and one year of
geometry. If a student needs help with algebra, the student
needs to take the appropriate steps to get help. There is no time during the
class to re-teach algebra.
Examples, exercises and activities provide a real-life context to help
students grasp mathematical concepts. Technology is utilized throughout the
course.
These are the topics we will cover in Pre-Calculus as determined by the Duval
County Public Schools district.
1. Polynomial and Rational Functions
2. Exponential and Logarithmic Functions
3. Basic Conics
4. Circular and Trigonometric Functions
5. Trigonometric Identities
6. Oblique Triangles
7. Vectors
8. Parametric and Polar Equations
9. Sequences and Series |
Getting
Assistance
When
Get help as soon
as you need it. Don't wait until the test is near. The new
material builds on the previous sections, so anything you don't understand now
will make future material difficult to understand.
Use the
Resources You Have Available
Ask questions in
class. You get help and stay actively involved in class.
Visit the
Instructor's Office Hours. Instructors like to see students who want
to help themselves.
Ask friends,
members of your study group, or anyone else who can help. The
classmate who explains something to you learns just as much as you do, for
he/she must think carefully about how to explain the particular concept or
solution in a clear way. So don't be reluctant to ask a classmate.
All students
need help at some point, so be sure to get the help you need.
Asking Questions
Don't be afraid to
ask questions. Any question is better than no question at all (at
least your Instructor/tutor will know you are confused). But a good
question will allow your helper to quickly identify exactly what
you don't understand.
Not too helpful
comment: "I don't understand this section." The best you
can expect in reply to such a remark is a brief review of the section, and
this will likely overlook the particular thing(s) which you don't
understand.
Good comment: "I
don't understand why f(x + h) doesn't equal f(x) + f(h)." This
is a very specific remark that will get a very specific response and
hopefully clear up your difficulty.
Good question:
"How can you tell the difference between the equation of a circle and
the equation of a line?"
Okay question:
"How do you do #17?"
Better question:
"Can you show me how to set up #17?" (the Instructor can
let you try to finish the problem on your own), or "This is how I
tried to do #17. What went wrong?" The focus of attention
is on your thought process.
Right after you get
help with a problem, work another similar problem by yourself.
You Control the Help You
Get
Helpers should be coaches,
not crutches. They should encourage you, give you hints as you need
them, and sometimes show you how to do problems. But they should not,
nor be expected to, actually do the work you need to do. They are
there to help you figure out how to learn math for yourself.
When you go to office
hours, your study group or a tutor, have a specific list of questions
prepared in advance. You should run the session as much as
possible.
Do you allow yourself
to become dependent on a tutor. The tutor cannot take the exams for
you. You must take care to be the one in control of tutoring
sessions.
You must recognize that
sometimes you need some coaching to help you through, and it is up to you
to seek out that coaching. |
We've heard you say that all students are not created equal when it comes to reasoning and math skills. Furthermore, we share your belief that the ability to reason in an organized and mathematically correct manner is essential to solving problems. That's why helping students improve their reasoning skills is also one of Cutnell & Johnson's primary goals. The following features will help students improve their reasoning skills:
Video Help, available through WileyPLUS, provides 3–5 minute office hour style videos tailored to the more challenging problems that bring together two or more physics concepts. These videos do not solve the problems, rather they point the student in the right direction by using a proven problem–solving technique: 1. Visualize the problem 2. Organize the data 3. Develop a reasoning strategy.
Math Skills appears as a sidebar throughout the text. It is designed to provide additional help with mathematics for students who need it, yet be unobtrusive for students who don't.
There's also a math skills module in WileyPLUS (a chapter 0) for students who want even more help.
Explicit reasoning steps in all examples explain what motivates the procedure for solving the problem before any algebraic or numerical work is done.
Reasoning Strategies for solving certain classes of problems are called out to encourage frequent review of the techniques used and help students focus on the related concepts
Analyzing Multiple–Concept Problems prompt students to combine one or more physics concepts before reaching a solution. First, they must identify the physics concepts involved in the problem, then associate each concept with an appropriate mathematical equation, and assemble the equations to produce a unified algebraic solution.
In order to reduce a complex problem into a sum of simpler parts, each Multiple-Concept example consists of four sections: Reasoning, Knowns and Unknowns, Modeling the Problem, and Solution.
Homework problems with associated Guided Online (GO) Tutorials have increased by 45% in this edition. Each of these problems in WileyPLUS includes a guided tutorial option (not graded) that instructors can make available for student access with or without penalty.
* GO tutorials facilitate strong problem–solving skills by providing a step by step guide on how to approach a problem. Multiple–choice questions in the GO tutorial include extensive feedback for both correct and incorrect answers. These multiple–choice questions guide students to the proper conceptual basis for the problem. The GO tutorial also includes calculational steps
Interactive LearningWare, available in WileyPLUS, consists of interactive examples, presented in a five-step format, designed to help improve each student's problem–solving skills.
Interactive Solutions, available in WileyPLUS, enable students to work out problems in an interactive manner while providing a model for the corresponding homework problem. |
D Martin, Glasgow University, UK
This blend of local coordinate methods and intrinsic differential geometry enables workers to read and do calculations in relativity and high energy particle research. It provides foundations for study in gauge theory, differential geometry and differential topology. Mathematical Reviews
Dr Martin's very readable differential geometry text for graduate students in physics could also be used for
independent study. American Mathematical Monthly
- provides a comprehensive account of basic manifold theory for post-graduate students - introduces the basic theory of differential geometry to students in theoretical physics and mathematics - contains more than 130 exercises, with helpful hints and solutions
This account of basic manifold theory and global analysis, based on senior undergraduate and post-graduate courses at Glasgow University for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology. Topology is included in two appendices because many courses on mathematics for physics students do not include this subject. |
Customer Reviews for New Leaf Publishing Group Exploring The World Of Mathematics
Show your students that numbers don't have to be difficult---in fact, they can be enjoyable! More than just another textbook, this supplement to your curriculum traces the history of mathematics principles and theory; features simple algebra, geometry, and scientific computations; and offers practical tips for everyday math use. Includes biblical examples, fun activities, chapter tests, and lots of illustrations and diagrams. 160 pages, softcover from New Leaf The World Of Mathematics
Review 1 for Exploring The World Of Mathematics
Overall Rating:
5out of5
Engaging
This book gives a history of the development of mathematics with some instruction on how to do the various types of math worked in. The text was engaging and easy to understand. Much of the book was suitable for middle schoolers, though some chapters were more high school level. There were useful black and white charts and illustrations. At the end of each chapter, there were 10 questions--most tested if you learned the important points in the chapter, but some were math problems based on what was learned. The answers were in the back.
The book occasionally referred to things in the Bible, like explaining the cubit as an ancient measurement of length. The author had math start with the ancient Egyptians (since, according to him, it wasn't needed before then because people were roaming herders).
Chapter 1 talked about ancient calendars and how the modern calendar was developed. Chapter 2 talked about marking the passage of time. Chapter 3 talked about the development of weights and measures from ancient ones to modern non-metric systems. Chapter 4 talked about the development of the metric system.
Chapter 10 talked about algebra and analytical geometry. Chapter 11 talked about network design, combinations & permutations, factorials, Pascal's triangle, and probability. Chapter 12 talked about the development of counting machines, from early mechanical calculators to modern digital calculators. Chapter 13 talked about the development of modern computers. Chapter 14 gave some math tricks and puzzles.
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Review 2 for Exploring The World Of Mathematics
Overall Rating:
5out of5
Date:November 11, 2007
Cindy Downes
Because I was taking College Algebra last semester, I picked up the book, Exploring the World of Mathematics, to read in order to supplement my understanding of math. Great choice! Not only did I learn more about mathematic principles but I learned more about the history of math, how math applies to everyday life, and even how math is used in scriptures!I suggest that sometime during your child's 5th-8th grade years, you go through each chapter with him - maybe as a summer course or one day a week on Friday. Most kids will like the book, too, as it teaches them how to solve logic problems that can fool their friends! Like this one: Have your friend secretly choose a number from one to ten. Tell him to add six to the number, double the results, and divide his answer by four. Next subtract half of the original number. When he is done, you can tell him what his number is 100% of the time. You'll have to read the book to find out how! |
Middle School Advanced Course: A Model One-Year Middle School Program
The content material described in this course contains the foundational understandings necessary to prepare students to pursue the study of algebra and geometry at a high school level, whether that content is encountered in traditionally organized or integrated courses.
It is designed to build on a rigorous K–6 experience such as one indicated by the expectations outlined in Achieve's Elementary Mathematics Benchmarks, Grades K–6 document or documents developed by the National Council of Teachers of Mathematics. It also builds on the National Assessment of Educational Progress (NAEP) elementary guidelines as well as the expectations set in many states' elementary grade standards. In particular, it is expected that students will come to this course with a strong conceptual foundation as well as computational facility with whole and rational numbers.
The mastery of this material in the single year-long middle school course described here can be accomplished only by very able and dedicated students. Students without the prerequisite knowledge outlined in Appendix A should instead be presented with this material in a two-year course sequence like that outlined in Achieve's model Middle School Course 1 and Middle School Course 2. During this Middle School Advanced Course, students will extend their understanding of the operations and properties of the rational number system to real numbers, including numbers expressed using exponents and roots. Important middle school concepts of ratio, rates, scaling, and similarity are studied. Algebraic expressions and relationships grow naturally out of work with numerical operations. Facility with variable expressions and numerical relationships expressed algebraically and graphically is critical to later success in more advanced mathematics courses. A major emphasis in this one-year course is on linear relationships; these are used to introduce students to the concept of a function and its multiple representations. Solving linear equations and connecting their solutions to the graph of the related linear function and to a contextual situation from which the equation might have arisen further prepare students for more rigorous mathematical modeling in later courses. This knowledge and skill also open the door to interesting and varied applications of the mathematics students are learning. Work with data analysis extends the algebraic lessons further into real life situations. Plane geometry as it relates to transformations and the geometry of circles as well as the in-depth study of slope will afford students opportunities to connect different branches of mathematics. Simple logical arguments that both verify and establish facts about geometric figures help solidify that knowledge. This Middle School Advanced Course also includes some basic work with probability. Two optional units are included that, while interesting and appropriate, may not be feasible within the time available in a typical 180-day school year. Upon completion of this course and its prerequisite materials, students should be prepared for success on eighth-grade state tests as well as the NAEP grade 8 assessment. They should also be prepared to successfully tackle algebra and geometry taught at the high school level.
Appropriate use of technology is expected in all work. In middle school this includes employing technological tools to assist students in creating graphs and data displays, transforming graphs, conceptualizing and analyzing geometric situations, and solving problems. Testing with and without technological tools is recommended.
How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of this middle school course to concepts students have encountered in earlier grades as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.
The Major Concepts below provide the focus for this one-year Middle School Advanced Course, which is intended to prepare students for a high school–level course in Algebra I or the first course in an Integrated Mathematics sequence. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.
Real Numbers, Exponents, Roots, and the Pythagorean Theorem
Variables and Expressions
Functions
Equations and Identities
Geometric Reasoning, Representation, and Transformations
Circles
Ratios, Rates, Scaling, and Similarity
Probability
Question Formulation and Data Collection
Number Bases [OPTIONAL ENRICHMENT UNIT]
Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for this Middle School Advanced Course.
Rational Number Operations
Numeric Relationships
Estimation and Approximation
Integer Exponents
Graphing in the Coordinate Plane
Systems of Measurement
Geometric Shapes and Measurement
Perimeter, Area, Volume, and Surface Area
Lines, Angles, and Triangles
Elementary Data Analysis and Probability
A. Real Numbers, Exponents and Roots and the Pythagorean Theorem
Students extend the properties of computation with rational numbers to real number computation, categorize real numbers as either rational or irrational, and locate real numbers on the number line. Powers and roots are studied along with the Pythagorean theorem and its converse, a critical concept in its own right as well as a context in which numbers expressed using powers and roots arise. Students apply this knowledge to solve problems.
Successful students will:
A1 Use the definition of a root of a number to explain the relationship of powers and roots.
If an = b, for an integer n ≥ 0, then a is said to be an nth root of b. When n is even and b > 0, we identify the unique a > 0 as the principal nth root of b, written .
Use and interpret the symbols and ; informally explain why , when .
By convention, for is used to represent the non-negative square root of a.
Estimate square and cube roots and use calculators to find good approximations.
Make or refine an estimate for a square root using the fact that if 0 ≤ a < n < b, then ; make or refine an estimate for a cube root using the fact that if a < n < b, then .
A2 Categorize real numbers as either rational or irrational and know that, by definition, these are the only two possibilities; extend the properties of computation with rational numbers to real number computation.
Approximately locate any real number on the number line.
Apply the definition of irrational number to identify examples and recognize approximations.
Square roots, cube roots, and nth roots of whole numbers that are not respectively squares, cubes, and nth powers of whole numbers provide the most common examples of irrational numbers. Pi (π) is another commonly cited irrational number.
Know that the decimal expansion of a rational number eventually repeats, perhaps ending in repeating zeros; use this to identify the decimal expansion of an irrational number as one that never ends and never repeats.
Recognize and use 22/7 and 3.14 as approximations for the irrational number represented by pi (π).
Determine whether the square, cube, and nth roots of integers are integral or irrational when such roots are real numbers.
A3 Interpret and prove the Pythagorean theorem and its converse; apply the Pythagorean theorem and its converse to solve problems.
Determine distances between points in the Cartesian coordinate plane and relate the Pythagorean theorem to this process.
B. Variables and Expressions
In middle school, students work more with symbolic algebra than in the previous grades. Students develop an understanding of the different uses for variables, analyze mathematical situations and structures using algebraic expressions, determine if expressions are equivalent, and identify single-variable expressions as linear or non-linear.
Successful students will:
B1 Interpret and compare the different uses of variables and describe patterns, properties of numbers, formulas, and equations using variables.
While a variable has several distinct uses in mathematics, it is fundamentally just a number we either do not know yet or do not want to specify.
Compare the different uses of variables.
Examples: When a + b = b + a is used to state the commutative property for addition, the variables a and b represent all real numbers; the variable a in the equation 3a - 7 = 8 is a temporary placeholder for the one number, 5, that will make the equation true; the symbols C and r refer to specific attributes of a circle in the formula C = 2πr; the variable m in the slope-intercept form of the line, y = mx + b, serves as a parameter describing the slope of the line.
Express patterns, properties, formulas, and equations using and defining variables appropriately for each case.
Two algebraic expressions are equivalent if they yield the same result for every value of the variables in them. Great care must be taken to demonstrate that, in general, a finite number of instances is not sufficient to demonstrate equivalence.
Analyze expressions to identify when an expression is the sum of two or more simpler expressions (called terms) or the product of two or more simpler expressions (called factors). Analyze the structure of an algebraic expression and identify the resulting characteristics.
Identify single-variable expressions as linear or non-linear.
Evaluate a variety of algebraic expressions at specified values of their variables.
Algebraic expressions to be evaluated include polynomial and rational expressions as well as those involving radicals and absolute value.
Use commutative, associative, and distributive properties of number operations to transform simple expressions into equivalent forms in order to collect like terms or to reveal or emphasize a particular characteristic.
Rewrite linear expressions in the form ax + b for constants a and b.
Choose different but equivalent expressions for the same quantity that are useful in different contexts.
Example: p + 0.07p shows the breakdown of the cost of an item into the price p and the tax of 7%, whereas (1.07)p is a useful equivalent form for calculating the total cost.
Demonstrate equivalence through algebraic transformations or show that expressions are not equivalent by evaluating them at the same value(s) to get different results.
Know that if each expression is set equal to y and the graph of all ordered pairs that satisfy one of these new equations is identical to the graph of all ordered pairs that satisfy the other, then the expressions are equivalent.
C. Functions
Middle school students increase their experience with functional relationships and begin to express and understand them in more formal ways. They distinguish between relations and functions and convert flexibly among the various representations of tables, symbolic rules, verbal descriptions, and graphs. A major focus at this level is on linear functions, recognizing linear situations in context, describing aspects of linear functions such as slope as a constant rate of change, identifying x- and y-intercepts, and relating slope and intercepts to the original context of the problem.
Successful students will:
C1 Determine whether a relationship is or is not a function; represent and interpret functions using graphs, tables, words, and symbols.
In general, a function is a rule that assigns a single element of one set–the output set—to each element of another set—the input set. The set of all possible inputs is called the domain of the function, while the set of all outputs is called the range.
Identify the independent (input) and dependent (output) quantities/variables of a function.
Make tables of inputs x and outputs f(x) for a variety of rules that take numbers as inputs and produce numbers as outputs.
Define functions algebraically, e.g., g(x) = 3 + 2(x - x2).
Create the graph of a function f by plotting and connecting a sufficient number of ordered pairs (x, f(x)) in the coordinate plane.
Analyze and describe the behavior of a variety of simple functions using tables, graphs, and algebraic expressions.
Construct and interpret functions that describe simple problem situations using expressions, graphs, tables, and verbal descriptions and move flexibly among these multiple representations.
C2 Analyze and identify linear functions of one variable; know the definitions of x- and y-intercepts and slope, know how to find them and use them to solve problems.
A function exhibiting a rate of change (slope) that is constant is called a linear function. A constant rate of change means that for any pair of inputs x1 and x2, the ratio of the corresponding change in value f(x2) - f(x2) to the change in input x2 - x1 is constant (i.e., it does not depend on the inputs).
Explain why any function defined by a linear algebraic expression has a constant rate of change.
Explain why the graph of a linear function defined for all real numbers is a straight line, identify its constant rate of change, and create the graph.
Determine whether the rate of change of a specific function is constant; use this to distinguish between linear and nonlinear functions.
Know that a line with a slope equal to zero is horizontal and represents a function, while the slope of a vertical line is undefined and cannot represent a function.
C3 Express a linear function in several different forms for different purposes.
Recognize that in the form f(x) = mx + b, m is the slope, or constant rate of change of the graph of f, that b is the y-intercept and that in many applications of linear functions, b defines the initial state of a situation; express a function in this form when this information is given or needed.
Recognize that in the form f(x) = m(x - x0) + y0, the graph of f(x) passes through the point (x0, y0); express a function in this form when this information is given or needed.
C4 Recognize contexts in which linear models are appropriate; determine and interpret linear models that describe linear phenomena; express a linear situation in terms of a linear function f(x) = mx + b and interpret the slope (m) and the y-intercept (b) in terms of the original linear context.
Common examples of linear phenomena include distance traveled over time for objects traveling at constant speed; shipping costs under constant incremental cost per pound; conversion of measurement units (e.g., pounds to kilograms or degrees Celsius to degrees Fahrenheit); cost of gas in relation to gallons used; the height and weight of a stack of identical chairs.
C5 Recognize, graph, and use direct proportional relationships.
Show that the graph of a direct proportional relationship is a line that passes through the origin (0, 0) whose slope is the constant of proportionality.
Compare and constrast the graphs of x = k, y = k, and y = kx, where k is a constant.
D. Equations and Identities
In this middle school course, students begin the formal study of equations. They solve linear equations and solve and graph linear inequalities in one variable. They graph equations in two variables, relating features of the graphs to the related single-variable equations. Solving systems of two linear equations in two variables graphically and understanding what it means to be a solution of such a system is also included in this unit. Interwoven with the development of these skills, students use linear equations, inequalities, and systems of linear equations to solve problems in context and interpret the solutions and graphical representations in terms of the original problem.
Successful students will:
D1 Distinguish among an equation, an expression, and a function; interpret identities as a special type of equation and identify their key characteristics.
An identity is an equation for which all values of the variables are solutions. Although an identity is a special type of equation, there is a difference in practice between the methods for solving equations that have a small number of solutions and methods for proving identities. For example, (x+2)2 = x2 + 4x + 4 is an identity which can be proved by using the distributive property, whereas (x+2)2 = x2 + 3x + 4 is an equation that can be solved by collecting all terms on one side.
Know that solving an equation means finding all its solutions and predict the number of solutions that should be expected for various simple equations and identities.
Explain why solutions to the equation f(x) = g(x) are the x-values (abscissas) of the set of points in the intersection of the graphs of the functions f(x) and g(x).
Recognize that f(x) = 0 is a special case of the equation f(x) = g(x) and solve the equation f(x) = 0 by finding all values of x for which f(x) = 0.
The solutions to the equation f(x) = 0 are called roots of the equation or zeros of the function. They are the values of x where the graph of the function f crosses the x-axis. In the special case where f(x) equals 0 for all values of x, f(x) = 0 represents a constant function where all elements of the domain are zeros of the function.
Use identities to transform expressions.
D2 Solve linear equations and solve and graph the solution of linear inequalities in one variable.
Common problems are those that involve break-even time, time/rate/distance, percentage increase or decrease, ratio and proportion.
Solve equations using the facts that equals added to equals are equal and that equals multiplied by equals are equal; more formally, if A = B and C = D, then A + C = B + D and AC = BD; use the fact that a linear expression ax + b is formed using the operations of multiplication by a constant followed by addition to solve an equation ax + b = 0 by reversing these steps.
Be alert to anomalies caused by dividing by 0 (which is undefined), or by multiplying both sides by 0 (which will produce equality even when things were originally unequal).
Graph a linear inequality in one variable and explain why the graph is always a half-line (open or closed); know that the solution set of a linear inequality in one variable is infinite, and contrast this with the solution set of a linear equation in one variable.
Explain why, when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed, but that when all other basic operations involving non-zero numbers are applied to both sides, the direction of the inequality is preserved.
D3 Recognize, represent, and solve problems that can be modeled using linear equations in two variables and interpret the solution(s) in terms of the context of the problem.
Rewrite a linear equation in two variables in any of three forms: ax + by = c, ax + by + c = 0, or y = mx + b; select a form depending upon how the equation is to be used.
Know that the graph of a linear equation in two variables consists of all points (x, y) in the coordinate plane that satisfy the equation and explain why, when x can be any real number, such graphs are straight lines.
Identify the relationship between linear functions in one variable, x maps to f(x) and linear equations in two variables f(x) = y or f(x) - y = 0; explain why the solution to an equation in standard (or polynomial) form (ax + b = 0) will be the point where the graph of f(x) = ax + b crosses the x-axis.
Identify the solution of an equation that is in the form f(x) = g(x) and relate the solution to the x-value (abscissa) of the point at which the graphs of the functions f(x) and g(x) intersect.
Know that pairs of non-vertical lines have the same slope if and only if they are parallel (or the same line) and slopes that are negative reciprocals if and only if they are perpendicular; apply these relationships to analyze and represent equations.
Represent linear relationships using tables, graphs, verbal statements, and symbolic forms; translate among these forms to extract information about the relationship.
D4 Determine the solution to application problems modeled by two linear equations and interpret the solution set in terms of the situation.
Determine either through graphical methods or comparing slopes whether a system of two linear equations has one solution, no solutions, or infinitely many solutions, and know that these are the only possibilities.
Represent the graphs of two linear equations as two intersecting lines when there is one solution, parallel lines when there is no solution, and the same line when there are infinitely many solutions.
Use the graph of two linear equations in two variables to suggest solution(s).
Since the solution is a set of ordered pairs that satisfy the equations, it follows that these ordered pairs must lie on the graph of each of the equations in the system; the point(s) of intersection of the graphs is (are) the solution(s) to the system of equations.
Recognize and solve problems that can be modeled using two linear equations in two variables.
Examples: Break-even problems, such as those comparing costs of two services
E. Geometric Representation and Transformations
Coordinate geometry affords middle school students the opportunity to make valuable connections between algebra concepts and geometry representations such as slope and distance. Students extend their elementary school experiences with transformations as specific motions in two-dimensions to transformations of figures in the coordinate plane. They describe the characteristics of transformations that preserve distance, relating them to congruence.
Successful students will:
E1 Represent and explain the effect of translations, rotations, and reflections of objects in the coordinate plane.
Identify certain transformations (translations, rotations and reflections) of objects in the plane as rigid motions and describe their characteristics; know that they preserve distance in the plane.
Demonstrate the meaning and results of the translation, rotation, and reflection of an object through drawings and experiments.
Identify corresponding sides and angles between objects and their images after a rigid transformation.
Show how any rigid motion of a figure in the plane can be accomplished through a sequence of translations, rotations, and reflections.
E2 Represent and interpret points, lines, and two-dimensional geometric objects in a coordinate plane; calculate the slope of a line in a coordinate plane.
Determine the area of polygons in the coordinate plane.
Know how the word slope is used in common non-mathematical contexts, give physical examples of slope, and calculate slope for given examples.
Find the slopes of physical objects (roads, roofs, ramps, stairs) and express the answers as a decimal, ratio, or percent.
Interpret and describe the slope of parallel and perpendicular lines in a coordinate plane.
Show that the calculated slope of a line in a coordinate plane is the same no matter which two distinct points on the line one uses to calculate the slope.
Use coordinate geometry to determine the perpendicular bisector of a line segment.
F. Circles
The main focus of this unit is on the study of circles, the relationships among their parts, the development of the formulas for the area and circumference and methods for approximating π.
Successful students will:
F1 Identify and explain the relationships among the radius, diameter, circumference and area of a circle; know and apply formulas for the circumference and area of a circle, semicircle, and quarter-circle.
Identify the relationship between the circumference of a circle and its radius or diameter as a direct proportion and between the area of a circle and the square of its radius or the square of its diameter as a direct proportion.
Demonstrate why the formula for the area of a circle (radius times one-half of its circumference) is plausible and makes geometric sense.
Show that for any circle, the ratio of the circumference to the diameter is the same as the ratio of the area to the square of the radius and that these ratios are the same for different circles; identify the constant ratio A/r2 = ½Cr/r2 = C/2r = C/d as the number π and know that although the rational numbers 3.14, or are often used to approximate π, they are not the actual values of the irrational number π.
Identify and describe methods for approximating π.
G. Ratios, Rates, Scaling, and Similarity
In conjunction with the study of rational numbers, middle school students examine ratios, rates, and proportionality both procedurally and conceptually. Proportionality concepts connect many areas of the curriculum, number, similarity, scaling, slope, and probability and serve as a foundation for future mathematics study. Examining proportionality first with numbers then geometrically with similarity concepts and scaling begins to establish important understandings for more formal study of these concepts in high school algebra and geometry courses.
Successful students will:
G1 Use ratios, rates, and derived quantities to solve problems.
Interpret and apply measures of change such as percent change and rates of growth.
Calculate with quantities that are derived as ratios and products.
Examples: Interpret and apply ratio quantities including velocity and population density using units such as feet per second and people per square mile; interpret and apply product quantities including area, volume, energy, and work using units such as square meters, kilowatt hours, and person days.
Solve data problems using ratios, rates, and product quantities.
Create and interpret scale drawings as a tool for solving problems.
A scale drawing is a representation of a figure that multiplies all the distances between corresponding points by a fixed positive number called the scale factor
G2 Analyze and represent the effects of multiplying the linear dimensions of an object in the plane or in space by a constant scale factor, r.
Use ratios and proportional reasoning to apply a scale factor to a geometric object, a drawing, or a model, and analyze the effect.
Describe the effect of a scale factor r on length, area, and volume.
G3 Interpret the definition and characteristics of similarity for figures in the plane and apply to problem solving situations.
Informally, two geometric figures in the plane are similar if they have the same shape. More formally, having the same shape means that one figure can be transformed onto the other by applying a scale factor.
Apply similarity in practical situations; calculate the measures of corresponding parts of similar figures.
Use the concepts of similarity to create and interpret scale drawings.
H. Probability
Students have an opportunity in this unit to apply both their rational number and proportional reasoning skills to probability situations. Students use theoretical probability and proportions to predict outcomes of simple events. Frequency distributions are examined and created to analyze the likelihood of events. The Law of Large Numbers is used to link experimental and theoretical probabilities.
Successful students will:
H1 Describe the relationship between probability and relative frequency; use a probability distribution to assess the likelihood of the occurrence of an event.
Recognize and use relative frequency as an estimate for probability.
If an action is repeated n times and a certain event occurs b times, the ratio b/n is called the relative frequency of the event occurring.
Use theoretical probability, where possible, to determine the most likely result if an experiment is repeated a large number of times.
Identify, create, and describe the key characteristics of frequency distributions of discrete and continuous data.
A frequency distribution shows the number of observations falling into each of several ranges of values; if the percentage of observations is shown, the distribution is called a relative frequency distribution. Both frequency and relative frequency distributions are portrayed through tables, histograms, or broken-line graphs.
Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drink. Based on these data, what is the best estimate of how many of the 2,352 students in the school could distinguish between the soft drinks?
Explain how the Law of Large Numbers explains the relationship between experimental and theoretical probabilities.
The Law of Large Numbers indicates that if an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions approaches p as the number of repetitions becomes arbitrarily large.
Use simulations to estimate probabilities.
Compute and graph cumulative frequencies.
I. Question Formulation and Data Collection
Students learn to design a study to answer a question; collect, organize, and summarize data; communicate the results; and make decisions about the findings. Technology is utilized both to analyze and display data. Students expand their repertoire of graphs and statistical measures and begin the use of random sampling in sample surveys. They assess the role of random assignment in experiments. They look critically at data studies and reports for possible sources of bias or misrepresentation. Students are able to use their knowledge of slope to analyze lines of best fit in scatter plots and make predictions from the data further connecting their algebra and data knowledge.
Successful students will:
I1 Formulate questions about a phenomenon of interest that can be answered with data; design a plan to collect appropriate data; collect and record data; display data using tables, charts, or graphs; evaluate the accuracy of the data.
Recognize the need for data; understand that data are numbers in context (with units) and identify units. Define measurements that are relevant to the questions posed; organize written or computerized data records, making use of computerized spreadsheets.
Understand the differing roles of a census, a sample survey, an experiment, and an observational study.
Select a design appropriate to the questions posed.
Use random sampling in sample surveys and random assignment in experiments, introducing random sample as a "fair" way to select an unbiased sample.
Represent univariate data; make use of line plots (dot plots), stem-and-leaf plots, and histograms.
Represent bivariate data; make use of scatter plots.
Describe the shape, center, and spread of data distributions.
Example: A scatter plot used to represent bivariate data may have a linear shape; a trend line may pass through the mean of the x and y variables; its spread is shown by the vertical distances between the actual data points and the line.
Identify and explain misleading uses of data by considering the completeness and source of the data, the design of the study, and the way the data are analyzed and displayed.
Examples: Determine whether the height or area of a bar graph is being used to represent the data; evaluate whether the scales of a graph are consistent and appropriate or whether they are being adjusted to alter the visual information conveyed.
I3 Summarize, compare, and interpret data sets by using a variety of statistics. Use percentages and proportions (relative frequencies) to summarize univariate categorical data.
Use conditional (row or column) percentages and proportions to summarize bivariate categorial data.
Use measures of center (mean and median) and measures of spread (percentiles, quartiles, and interquartile range) to summarize univariate quantitative data.
Interpret the slope of a linear trend line in terms of the data being studied.
Use box plots to compare key features of quantitative data distributions.
I4 Read, interpret, interpolate, and judiciously extrapolate from graphs and tables and communicate the results.
State conclusions in terms of the question(s) being investigated.
Use appropriate statistical language when reporting on plausible answers that go beyond the data actually observed.
Use oral, written, graphic, pictorial and multi-media methods to create and present manuals and reports.
I5 Determine whether a scatter plot suggests a linear trend.
Visually determine a line of good fit to estimate the relationship in bivariate data that suggests a linear trend.
Identify criteria that might be used to assess how good the fit is.
The following unit is an extension or enrichment unit, which while interesting and appropriate, may not be feasible time-wise in a traditional 180-day school year.
J. Number Bases [OPTIONAL ENRICHMENT UNIT]
This should be used as an optional unit of study if time permits. Using their understanding of the base-10 number system, students represent numbers in other bases. Computers and computer graphics have made much more important the knowledge of how to work with different base systems, particularly binary.
Successful students will:
J1 Identify key characteristics of the base-10 number system and adapt them to the binary number base system.
Represent and interpret numbers in the binary number system.
Apply the concept of base-10 place value to understand representation of numbers in other bases.
Example: In the base-8 number system, the 5 in the number 57,273 represents 5 x 84.
Convert binary to decimal and vice versa.
Encode data and record measurements of information capacity using the binary number base system.
Appendix A: Prior Knowledge
The following expectations, which are included in the model two-year middle school course sequence (Middle Course 1 and Middle School Course 2), are essential prerequisites for success in the one-year middle school program (Middle School Advanced Course). Students must develop proficiency in these expectations prior to embarking upon this one-year advanced course. This means that schools opting for this one-year Middle School Advanced Course must adjust the mathematics curriculum in earlier grades to include these expectations.
PK.A. Number Representation and Computation
Successful students will:
PK.A1 Extend and apply understanding about rational numbers; translate among different representations of rational numbers.
Rational numbers are those that can be expressed in the form where p and q are integers and q ≠ 0.
Use inequalities to compare rational numbers and locate them on the number line; apply basic rules of inequalities to transform numeric expressions involving rational numbers.
Demonstrate understanding of the algorithms for addition, subtraction, multiplication, and division (non-zero divisor) of numbers expressed as fractions, terminating decimals, or repeating decimals by applying the algorithms and explaining why they work.
A pattern is a sequence of numbers or objects constructed using a simple rule. Of special interest are arithmetic sequences, those generated by repeated addition of a fixed number, and geometric sequences, those generated by repeated multiplication by a fixed number.
PK.A4 Know and apply the Fundamental Theorem of Arithmetic.
Every positive integer is either prime itself or can be written as a unique product of primes (ignoring order).
Identify prime numbers; describe the difference between prime and composite numbers; determine and divisibility rules (2, 3, 5, 9, 10), explain why they work, and use them to help factor composite numbers.
Determine the greatest common divisor and least common multiple of two whole numbers from their prime factorizations; explain the meaning of the greatest common divisor (greatest common factor) and the least common multiple and use them in operations with fractions.
Use greatest common divisors to reduce fractions and ratios n:m to an equivalent form in which the gcd (n, m) = 1.
Fractions in which gcd (n, m) = 1 are said to be in lowest terms.
Write equivalent fractions by multiplying both numerator and denominator by the same non-zero whole number or dividing by common factors in the numerator and denominator.
Add and subtract fractions by using the least common multiple (or any common multiple) of denominators.
PK.A5 Identify situations where estimates are appropriate and use estimates to predict results and verify the reasonableness of calculated answers.
Use rounding, regrouping, percentages, proportionality, and ratios as tools for mental estimation.
Develop, apply, and explain different estimation strategies for a variety of common arithmetic problems.
Explain the phenomenon of rounding error, identify examples, and, where possible, compensate for inaccuracies it introduces.
Examples: Analyzing apportionment in the U.S. House of Representatives; creating data tables that sum properly; analyzing what happens to the sum if you always round down when summing 100 terms.
PK.A6 Use the rules of exponents to simplify and evaluate expressions.
Evaluate expressions involving whole number exponents and interpret such exponents in terms of repeated multiplication.
PK.A7 Know and apply the definition of absolute value.
The absolute value is defined by |a| = a > 0 and |a| = -a if a < 0.
Interpret absolute value as distance from zero.
Interpret absolute value of a difference as "distance between" on the number line.
PK.A8 Analyze and apply simple algorithms.
Identify and give examples of simple algorithms.
An algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task that, given an initial state, will terminate in a well-defined end-state. Recipes and assembly instructions are everyday examples of algorithms.
Analyze and compare simple computational algorithms.
Examples: Write the prime factorization for a large composite number; determine the least common multiple for two positive integers; identify and compare mental strategies for computing the total cost of several objects.
Analyze and apply the iterative steps in standard base-10 algorithms for addition and multiplication of numbers.
PK.B. Measurement Systems
Successful students will:
PK.B1 Make, record, and interpret measurements.
Recognize that measurements of physical quantities must include the unit of measurement, that most measurements permit a variety of appropriate units, and that the numerical value of a measurement depends on the choice of unit; apply these ideas when making measurements.
Recognize that real-world measurements are approximations; identify appropriate instruments and units for a given measurement situation, taking into account the precision of the measurement desired.
Plan and carry out both direct and indirect measurements.
Indirect measurements are those that are calculated based on actual recorded measurements.
Apply units of measure in expressions, equations, and problem situations; when necessary, convert measurements from one unit to another within the same system.
Use measures of weight, money, time, information, and temperature; identify the name and definition of common units for each kind of measurement.
Record measurements to reasonable degrees of precision, using fractions and decimals as appropriate.
A measurement context often often defines a reasonable level of precision to which the result should be reported.
Example: The U.S. Census bureau reported a national population of 299,894,924 on its Population Clock in mid-October of 2006. Saying that the U.S. population is 3 hundred million (3x108) is accurate to the nearest million and exhibits one-digit precision. Although by the end of that month the population had surpassed 3 hundred million, 3x108 remained accurate to one-digit precision.
Calculate the perimeter and area of triangles, quadrilaterals, and shapes that can be decomposed into triangles and quadrilaterals that do not overlap; know and apply formulas for the area and perimeter of triangles and rectangles to derive similar formulas for parallelograms, rhombi, trapezoids, and kites.
Given the slant height, determine the surface area of right prisms and pyramids whose base(s) and sides are composed of rectangles and triangles; know and apply formulas for the surface area of right circular cylinders, right circular cones, and spheres; explain why the surface are of a right circular cylinder is a rectangle whose length is the circumference of the base of the cylinder and whose width is the height of the cylinder.
Given the slant height, determine the volume of right prisms, right pyramids, right circular cylinders, right circular cones, and spheres.
PK.C. Angles and Triangles
PK.C1 Know the definitions and properties of angles and triangles in the plane and use them to solve problems.
Know and apply the definitions and properties of complementary, supplementary,interior, and exterior angles.
Know and distinguish among the definitions and properties of vertical, adjacent, corresponding, and alternate interior angles; identify pairs of congruent angles and explain why they are congruent.
PK.C2 Know and verify basic theorems about angles and triangles.
Know the triangle inequality and verify it through measurement.
In words, the triangle inequality states that any side of a triangle is shorter than the sum of the other two sides; it can also be stated clearly in symbols: If a, b, and c are the lengths of three sides of a triangle, then a < b + c, b < a + c, and c < a + b.
Verify that the sum of the measures of the interior angles of a triangle is 180°.
Verify that each exterior angle of a triangle is equal to the sum of the opposite interior angles.
Show that the sum of the interior angles of an n-sided convex polygon is (n - 2) x 180°.
Explain why the sum of exterior angles of a convex polygon is 360°.
PK.D. 3-Dimensional Geometry
Successful students will:
PK.D1 Visualize solids and surfaces in three-dimensional space.
Relate a net, top-view, or side-view to a three-dimensional object that it might represent; visualize and be able to reproduce solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views).
Interpret the relative position and size of objects shown in a perspective drawing.
Visualize and describe three-dimensional shapes in different orientations; draw two-dimensional representations of three-dimensional objects by hand and using software; sketch two-dimensional representations of basic three-dimensional objects such as cubes, spheres, pyramids, and cones.
Create a net, top-view, or side-view of a three-dimensional object by hand or using software; visualize, describe, or sketch the cross-section of a solid cut by a plane that is parallel or perpendicular to a side or axis of symmetry of the solid.
PK.E. Data Analysis
For univariate data, make use of frequency and relative frequency tables and bar graphs; for bivariate data, make use of two-way frequency and relative frequency tables and bar graphs.
PK.F. Probability
Successful students will:
PK.F1 Represent probabilities using ratios and percents; use sample spaces to determine the (theoretical) probabilities of events; compare probabilities of two or more events and recognize when certain events are equally likely.
The odds of an event occurring is the ratio of the number of favorable outcomes to the number of unfavorable outcomes, whereas the probability is the ratio of favorable outcomes to the total number of possible outcomes. |
Graphical illustration of the Riemann sums of a function defined by its graph. The tool allows selecting the point inside the subintervals in several ways which helps show the dependence of the approx... More: lessons, discussions, ratings, reviews,...
Experiment with left, right, and midpoint Riemann sums for functions you enter, as well as inscribed rectangles, circumscribed rectangles, and the trapezoid rule. Includes extensive directions for cu... More: lessons, discussions, ratings, reviews,...
This activity builds on the previous activity, One Type of Integral, by suggesting two more efficient ways of estimating the area under a curve (definite integral) than counting squares: adding the ar... |
Lecture 40: Introduction to functions
Embed
Lecture Details :
An introduction to functions.
Course Description :
This is the original Algebra course on the Khan Academy and is where Sal continues to add videos that are not done for some other organization. It starts from very basic algebra and works its way through algebra II. |
Book Description: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. The 8th edition of Bluman provides a significant leap forward in terms of online course management with McGraw-Hill's new homework platform, Connect Statistics – Hosted by ALEKS. Statistic instructors served as digital contributors to choose the problems that will be available, authoring each algorithm and providing stepped out solutions that go into great detail and are focused on areas where students commonly make mistakes. From there, the ALEKS Corporation reviewed each algorithm to ensure accuracy. The result is an online homework platform that provides superior content and feedback, allowing students to effectively learn the material being taught.
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Math Helper Math Helper
the author of this app says:
MathHelper allows you to solve math exercises and get full steps of sollution
ATTENTION: Finally! Math Helper opearates on English and German language (including built-in math reference)!!! 2.0 version is released!
MathHelper is a unique application for students. The uniqueness of the application is that it allows us to see not only the answer, but a detailed solution of the problem. Now you do not need to order the work in math or ask for help from classmates - assistant in mathematics will do everything himself. In addition to solving problems, the application includes the theory on these topics and a scientific calculator.
Indispensable application for the pupils or students of the institutes, schools, during the session, exams, tests, and just learning, it is much better than any cheat sheets!
MathHelper Lite allows you to solve a wide range of tasks. MathHelper allows you to quickly solve typical mathematical exercises of the linear and vector algebra, devided in 4 sections:
3. Vector Algebra - Vectors: *Finding the magnitude or length of a Vector *Collinearity of two Vectors *Orthogonality of vectors *Vector Addition, Subtraction and Scalar Multiplication *Vector Multiplication *Finding the Angle Between two Vectors *Finding the cosine of the angle between the vectors AB and AC *Finding the projection of one Vector on another *Coplanarity of the Vectors
4. Vector Algebra - Figures: *Calculate the area of a triangle *Whether the four points lie on one plane *Calculate the volume of a tetrahedron (pyramid) *Find the volume and height of a tetrahedron (pyramid) Now it's operating in German and is extended in the following areas and tasks:
5 Probability Theory (new in v.2!):
*Finding the expected value *Finding the sampe variance *Finding the number of permutations of n distinct objects *Finding The number of permutations of n objects taken k at a time
6 The number sequences (new in v.2!):
*Prime Factorization *Finding LCM and GCD *The least common multiple and greatest common divisor *Complex numbers: Addition, subtraction, multiplication and division *Erection of a complex number to a power *The arithmetic and geometric progressions: n-th term and the sum of first n terms of the progression *Calculating the Fibonacci numbers *Solving Diophantine equations *Finding the value of the Euler's totient function *Finding Factorial of n! |
Short description
Classroom Mathematics is the mathematics teacher's choice for over 21 years.
Teachers feel confident in the proven success of the Classroom Mathematics method:
·Classroom Mathematics has carefully graded exercises in the Learners' Book.
·Classroom Mathematics has worked example answers in the Teacher's Guide.
·Classroom Mathematics covers all mathematical concepts thoroughly.
·Classroom Mathematics prepares learners for continuous assessment.
This new edition of the popular Classroom Mathematics course meets the requirements of the Revised National Curriculum Statement. The five Learning Outcomes and all the Assessment Standards for Grade 7 Mathematics are covered.
The Classroom MathematicsPractice Book contains extra revision exercises to provide more drill and practice in all the Learning Outcomes. This course is also available in Afrikaans as Wiskunde vir die Klaskamer. |
Guided Tour of Mathematical Methods For the Physical Sciences
9780521834926
ISBN:
0521834929
Edition: 2 Pub Date: 2004 Publisher: Cambridge Univ Pr
Summary: Mathematical methods are essential tools for all physical scientists. This textbook provides a comprehensive guided tour of the mathematical knowledge and techniques needed by students in this area. |
Mathematics 308 - Fall 1997 - Euclidean geometry
Fall term
Section 101
1:30-2:30 M W F
Buchanan A205
In the course this fall, as in previous years,
I will show how mathematics and
computers can be used together to produce
graphics of mathematical interest.
The primary programming language to be
used is PostScript. It is ideal for this purpose
because its imaging model uses affine geometry in a crucial manner.
Towards the end of the course, elementary
3D graphics including perspective, and
perhaps something about the regular solids, will
be discussed. New this year:
I will also allow work in Java,
and will arrange weekly tutorials for those interested in it.
Towards the end of the term, students
will have to propose and carry out their own projects.
You can even look at
a previous year's projects.
Students will be
given accounts in the Mathematics Department undergraduate
computer laboratory, and will also be able to
run GhostScript or GhostView
on PC-compatible machines or Macintoshes elsewhere.
The documentation below is in both PostScript
and PDF formats.
For reading PostScript you can use
GhostView & GhostScript if you do not
have another PostScript file browser.
For PDF files
Acrobat Reader is available from Adobe.
We have PostScript help on our
local help facility
If you would like to know something that isn't there, ask about it.
A better error handler than
the default one that comes with Ghostscript.
It is the same as the one made by Adobe,
except that one line of optional output has been suppressed,
in order to make it more convenient.
To use this program, put (ehandler.ps) run at the beginning
of a program.
Homework
Approximating the graph of
cos(t) with Bezier curves. The picture has two pages, the
second at four times the scale
of the first. Note the different graphs drawn with different
numbers of Bezier curves: green = 2, blue = 4, black = 8.
(Of course this fine point makes the code very difficult to read.)
The picture Simplified source |
MDM4U Grade 12 Data Management Math Course Description
ThisI often receive emails from teachers across Ontario looking for some resources for the Grade 12 MDM4U Data Management course. I always loved teaching this course, but haven't had the opportunity in the past year. Take what you'd like from here. If it helps you along the way, I'd love to hear from you in the comments section at the bottom of the page.
MDM4U – Mathematics of Data Management – Grade 12 – Worksheets
McGraw-Hill Ryerson – Digital Textbook
McGraw-Hill Ryerson Textbook: Mathematics
of Data Management in PDF format.
Also included is the full solution manual for even numbered problems.Download
Adobe Reader to view files. All files are numbered
according to the order of the textbook, not our course! |
Course 1- make sure you study the notes given today and the notes in your notebook. Know the formulas found on the KNOW THESE paper in your notebook. Practice doing some of the perimeter and area problems on the review sheet. You should also know about PEMDAS, exponents, and perfect squares. Don't forget to do your graphs.
Course 2- make sure you review all about central tendency. You should know all about stem and leaves, box and whiskers, and frequency tables. You should also know about exponents, PEMDAS, prime factorization, and gcf. |
purpose of this book is to present material on elementary statistical methods in a succinct manner, to extend the introductory ideas into analysis of variance and experimental design, and to explain without formal mathematical proof the assumptions on data necessary for the v ...
We use math every day, sometimes without even realizing it! Kid-friendly, real-life situations show readers how they can put math to work in their day-to-day activities. A variety of problem-solving activities and graphic organizers make these books ideal for young learners. . |
The Mathematics Curriculum: Mathematics Across the Curriculum of what will be needed for students studying other subjects, and where differences of approach in mathematics and science are identified the reasons behind the differences are elucidated. However, it emphasises that this book may help teachers, but cannot necessarily solve the problems, the best solution is communication between teachers.
Subjects surveyed include all the sciences, geography, economics and social studies, technical subjects, domestic subjects, physical education and Art |
Flossmoor PrecalculusArithmetic with Polynomials and Rational Expressions
Topics may include performing arithmetic operations on polynomials, understanding the relationship between zeros and factors of polynomials, using polynomial identities to solve problems, and rewriting rational expressions. Reasoning with Equ |
Course Listing
Math 1050 College Algebra
Description: This course explores the concept of functions: polynomial, rational, inverse, logarithmic and exponential; with an emphasis on graphing. Solving systems of equations using matrix methods is covered along with conic sections. Other topics may include sequences, mathematical induction and the binomial theorem. The course involves the extensive use of graphing calculators.
Credits: 4
Prerequisite(s): A recent (within the last two years) Math ACT of 23 or a grade of "C" or better in Math 1010 within the last two years. |
Features:
first part of book is a dictionary, giving brief and simple explanations of mathematical terms, often with examples and diagrams
second part of book contains a detailed reference section, providing an overview of key ideas about a wide range of mathematical topics, including tables, number systems, charts, 2-D shapes, 3-D shapes, measures, conversion tables, equivalences, formulas, rules, explanations and symbols |
In math, as in any form of communication, there are rules that are agreed upon so that everyone can understand exactly what is being communicated. Oral and written languages use vocabulary, grammar, and sentence structure to communicate effectively. Mathematics uses its own form of these entities as well. Learning the language of mathematics is a key aspect of understanding the concepts being communicated. Numbers (analogous in some ways to the alphabet in which a language is written) and how these numbers are combined (much as letters are combined in the spelling of words) are the foundation for the study of algebra. This chapter will introduce the basic building blocks of algebra and give you a sturdy foundation for your future study. |
Mathematics in Education
Turns your Sudden Motion Sensor-equipped laptop into a three-axis seismograph. Turns your Sudden Motion Sensor-equipped laptop into a three-axis seismograph. It shows a scrolling chart of the three axes of acceleration, reading up to five hundred...
Terrific Triangles is a math facts drill-and-practice program that teaches fact families for both addition/subtraction and multiplication/division. Terrific Triangles is a math facts drill-and-practice program that teaches fact families for both...
The award-winning periodic table of the elements for the Macintosh. The award-winning periodic table of the elements for the Macintosh. In addition to the usual information found in such programs, The Atomic Mac also contains a wealth of nuclear...
Solve mathematical models applied to technical problems of various type. Solve mathematical models applied to technical problems of various type. Documents can be realized and used as calculation models for a specific mathematical technical...
Do you have a homework assignment that needs the complete working out for a matrix question, and requires that it be neatly typed out? Do you have a homework assignment that needs the complete working out for a matrix question, and requires that...
Controls up to 4 USB or Firewire connected cameras during an eclipse so that you can be free to concentrate on observing the event visually. Controls up to 4 USB or Firewire connected cameras during an eclipse so that you can be free to...
Visualization tool for the N-body problem. Visualization tool for the N-body problem. The first program of its kind for Mac OS X, it has an intuitive interface, beautiful graphics and an accurate and fast core physics engine. Cavendish is named...
A Cocoa application dedicated to the processing of astronomical digital images taken through a telescope. A Cocoa application dedicated to the processing of astronomical digital images taken through a telescope. It is a a€sUniversal binarya€t...
Pythagorean Theorem is a text-based program that uses the formula A2 B2 = C2 to calculate the length of any side of a right triangle, provided you enter the other two. Pythagorean Theorem is a text-based program that uses the formula A2 B2 = C2 to...
Marketiva specializes in providing traders with high quality online trading services. Marketiva specializes in providing traders with high quality online trading services. With a team of dedicated financial specialists and technical support...
AlgeXpansion is an aplication that teaches algebraic expansion. AlgeXpansion is an aplication that teaches algebraic expansion. The program is capable of generating hundreds of sums for drills to ensure that the student masters the skills. It...
This program puts a set of problems on the screen. This program puts a set of problems on the screen. Each consists of two digits, and they are to be added or multiplied. There is no penalty for wrong answers. As soon as the user provides the...
Control engineers and instrumentation technicians require software tools to test communications both to and from Programmable Logic Controllers (PLCs), Remote Terminal Units (RTUs), and other logic solving devices. Control engineers and...
The best matrices calculator there is. The best matrices calculator there is.It is a calculator for real and complex matrices.It is not just a calculator you can write your own programs on it. Can do all manipulations on matrices. It can do add,...
A professional numerology decoding program that is easy to use. A professional numerology decoding program that is easy to use. It enables you to enter names, dates, letters, or numbers in any combination. Q-Decode will then break down the data...
Teaches the concepts of digital electronic circuits. Teaches the concepts of digital electronic circuits. The integrated schematic entry and simulation software was designed specifically for educational use and can be applied in minutes. Probes,...
An easy-to-use utility for backing up and restoring Garmin GPS waypoints and routes. An easy-to-use utility for backing up and restoring Garmin GPS waypoints and routes. All GPS waypoint and route data can be completely restored or you can select...
A word-processor-like editor specifically designed for use in high school and college-level algebra-based physics courses. A word-processor-like editor specifically designed for use in high school and college-level algebra-based physics courses.... |
A graphing calculator is required for this course. No particular model is required, but theTI-89 and/or TI-83 will be used for demonstrations.
It is advisable to have a notebook in which to keep homework assignments, class notes, handouts, and returned quizzes.
A pencil is the preferred writing instrument. You may take class notes and do homework with an instrument of your choosing, but it is expected that you will do all tests and quizzes in pencil.
Classwork & Homework
In order to master most mathematical concepts or processes, practice is required. Homework is assigned almost every night to give you opportunities to practice. You should do the homework with the idea of engaging the concepts and doing the assignment well, not just to get it done. You should keep your homework and class notes in a notebook of some kind. Organization is one of the keys to success for a student.
You will be divided into groups, and your first task each day is to check homework within your group. We will discuss any problems that cannot be done by anyone in a group. While you are checking your homework within your group, I will visit each group to see that you have done your work. Each time you haven't completed your assignment, a zero (0) will be recorded. Beginning with the third zero and for every zero thereafter, one (1) point will be deducted from your quarter average.
At times, I may take up your homework and grade it. I sometimes take a grade on your homework by giving a short quiz asking you to copy your solutions to three or four problems from the assignment.
Grading
Daily work and major tests are factors used in computing a grade for each quarter. Class work, homework, quizzes, and other types of daily (or short term) activities are used to determine the daily average for the quarter. Two to four major tests will be given during the quarter, and each will weigh equally in determining a test average for the quarter. The quarter grade is computed using the formula:
Qtr Grade = (2/3)*(test average) + (1/3)*(daily average)
Students are required to take a comprehensive examination at the end of each semester. The semester grade is computed from the quarter grades and the examination grade using the formula: |
The course presents to students knowledge on basic numerical methods:
matrix operations, solving systems of linear algebraic equations and regression. Another part of the lecture deals with polynomial interpolation and solution of one-dimensional nonlinear equations.
After successful passing of the course the students should be able to
- list and describe basic numerical methods lectured
- successfully apply these methods for solving a specified problem.
Syllabus
1) Number representation in a computer,precision, accuracy.
Errors in numerical algorithms,
propagation of the errors.
Stability of the algorthims.
Ill-posed methods. |
Tailored to both the specification and the tier, this Student Book delivers exactly what students and teachers need to cover the unit in exactly the right depth.
Synopsis:
* Supports teachers' understanding of AO2 and AO3 through clearly labelled AO2/3 questions in the exercises. * Packed with graded questions reflect the level of demand required, so students and teachers can see their progression. * Includes worked examples throughout the book to break the maths down into easy chunks. * Uses feedback to highlight common errors . |
To create mathematically, scientifically and technologically literate and functional learners who will be successful in a business world that relies on calculators, computers, scientific and mathematical procedures, rapidly growing and extensively applied in diverse situations. |
Originally posted by Speedbump If you have not done the infinite series/Fourier/Laplace math yet, then I'll bet that is what discrete math is. Don't worry, it would be easy if you are good at integration. Just some formulae to remember. I am not that great at integration, and managed to do the series stuff just fine.
You will use Fourier/Laplace a lot in "real life" engineering, but you won't have to do it the way they teach you in math classes. You'll use MathLab or some other program to do it all for you
I think Nebraska will still have a lot of work to do next year. I am not at all inpressed with Jamaal Lord at QB, and our defense has to get over losing Charlie McBride a few years ago and then Craig Bohl last year. At least Solich got smart and hired himself an offensive coordinator.
K State runs our offense better than we do right now. If Lord is our QB, and our defense isn't any better than last year, I say the sign stays Purple.
Integration? Can we quit with the integration already? I'm so sick of it . Actually I just use my TI-89, it does most of it for me. I just have to disguise it and make it look like I'm showing work come exam time . Integration is NOT my strongpoint LOL, there's a little too much guesswork in it. Memorizing formulas though, that I can do. Photographic memory... that's how I survive
Oh, and uh... MathLab is GOOD!!!
I personally hope the sign changes back to red. Not because I don't like my team, but because I think K-State purple is the UGLIEST color known to man. I like red though. My car is blood red, I like it so much
hm... I think there should be a Homework/School related Discussion Forum under AMDforum~@!!!
ok... i got 2 questions related to my I.T. assignment that I can't solve right now... I will keep reading my books but wanna know if you guys can help me a bit...
1. Consider the interrupt that occurs at the completion of a disk transfer. Describe the steps that take place after the interrupt occurs... (This is da question... I was wondering if Disk Transfer = DMA Transfer???)
2. Anyone here knows what is LMC (Little Man Computer)??? If one of you do know what i m talking about... then I will post the question here cause it's a long one.....
THX guys.... sorry to turn this thread into Personal/School-related thread AGAIN~!!! ... but i really need help~!@
for the disk transfer question.. i tried my best to B.S. a "sounds-like" answer and i hope it's correct....
for the "LMC" .. here's da question...
Suppose that the instruction format for a modified LMC requires 2 consecutive locations for each instruction. The high-order digits of the instruction are located in the first mail slot, followed by the lower-order digits. The IR (Instruction Register) is large enough to hold the entire instruction and can be addressed as IR [high] and IR [low] to load it. You may assume that the op code part of the instruction uses IR [high] and that the address is found in IR [low]. Write the fetch-execute cycle for an ADD instruction on the machine.
if anyone know what's LMC or know how's LMC function... plz teach me what is it... I rather you teach me the concept of it (i have a hard time how the register work...) than tell me a straight forward answer....
anyways... thx Speedbump and Sephiroth for trying to help...
I was wondering why da hell I (people in I.T.) have to leran this kinda stuff... isn't this related to Hardware/Software Enginner student???
The ol' softmodded 9500 got me 16,611 in 3dmark01 and 5,264 in 3dmark03 now that it has a decent chipset and proc to back it up.
I decided to post my most recent results with the 2 benches in my sig.
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This is a challenging and rigorous course offered to those students who have demonstrated an advanced proficiency in mathematics in sixth grade. The material covered in the accelerated course is presented at a faster pace with the expectation that the students have retained their advanced skills and can perform with greater proficiency and on their own.The accelerated course focuses on using the order of operation principles, solving equations and inequalities, applying rational numbers and integers to real life problems, relating rates, proportions, and percents, developing spatial thinking skills and exploring linear functions.A course grade of 85% in the 6th grade advanced course is recommended. |
The Math Forum Professional Development
The Math Forum is a community of mathematicians, teachers, and researchers working
together to improve math education. We recognize the opportunities and challenges
inherent in mathematics instruction and so, have developed courses and workshops to
support the teacher-student experience, many in conjunction with the Math Forum's popular
Problem of the Week.
The programs below are offered throughout the year and also may be customized to meet the
specific needs of educator groups. A complete schedule is available on the Math Forum's website.
Course Descriptions
The Math Forum's Problem-Solving Process
The course aligns well with the Math Forum's Problems of the Week but also could be used
to develop techniques to use with problem-solving prompts as well. Registration Information
Moving Students from Arithmetic to Algebra
This course examines a continuum of student work from the Math Forum's Problems of
the Week and how to move students' thinking along the continuum productively. Registration Information
Problem of the Week (PoW) Class Membership: Resources and Strategies for Effective Implementation
Designed for current PoW subscribers, this six-week course provides a basis of understanding allof the
features and resources associated with PoW membership. Registration Information
Problem Solving Strategies
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The book is intended to be an introductory text for mathematics and computer science students at the second and third year level in universities. It gives an introduction to the subject with sufficient theory for that level of student, with emphasis on algorithms and applications. |
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Math 250-CMATLAB Assignment #11 Revised 5/25/00LAB 1: Introduction to MATLABIn this lab you will learn how to use MATLAB to create and operate on matrices and vectors. The commands needed to do this are short and easy to remember, because MATL
Math 250-CMATLAB Assignment #31 Revised 5/24/00LAB 3: Complete Solution to Ax = bIn this lab you will use MATLAB to study the complete solution of the linear equation Ax = b, where A is a given m n matrix, b is a given m 1 column vector and
Math 250-CMATLAB Assignment #41 Revised 5/23/00LAB 4: Vector Spaces and Approximate Solutions to Ax = bIn this lab you will use MATLAB to study four topics: 1) The subspace spanned by a set of vectors, and the important concepts of independenc
Math 250-CMATLAB Assignment #51 Revised 5/23/00LAB 5: A = QR Factorization, Determinants, and Eigenvalues/EigenvectorsIn this lab you will use MATLAB to study three topics: 1) How to transform a given set of basis vectors into an orthonormal b
Math 250-CMATLAB Assignment #61 Revised 5/23/00LAB 6: Symmetric and Positive-Definite Matrices, Singular Value Decomposition, and Linear TransformationsIn this lab you will use MATLAB to transform a matrix into a diagonal matrix by finding its
Intro. to Linear Algebra 250-C Extra Credit Project 1 - Graphs and Matrices Please write all answers on separate sheets of paper. Your answers should be numbered and in the same order in which the problems appear. Your project should be stapled and y
Intro. to Linear Algebra 250-C Extra Credit Project 2 - Graphs and Markov Processes Introduction: In this project we will learn about the connection between linear algebra and market distribution. Suppose that, every day, there are three types of ent
Math 250-CMATLAB Assignment #11 Revised 9/15/02LAB 1: Matrix and Vector Computations in MATLABIn this lab you will use MATLAB to study the following topics: How to create matrices and vectors in MATLAB. The commands to do this are short and e
Math 250-CMATLAB Assignment #21 Revised 8/20/02LAB 2: Linear Equations and Matrix AlgebraIn this lab you will use MATLAB to study the following topics: Solving a system of linear equations by using the row reduced echelon form of the augmente
Math 250-CMATLAB Assignment #31 Revised 10/15/02LAB 3: LU Decomposition and DeterminantsIn this lab you will use MATLAB to study the following topics: The LU decomposition of an invertible square matrix A. How to use the LU decomposition to
Math 250-CMATLAB Assignment #41 Revised 10/25/02LAB 4: General Solution to Ax = bIn this lab you will use MATLAB to study the following topics: The column space Col(A) of a matrix A The null space Null(A) of a matrix A. Particular solutions
Math 250-CMATLAB Assignment #51 Revised 11/13/02LAB 5: Eigenvalues and EigenvectorsIn this lab you will use MATLAB to study these topics: The geometric meaning of eigenvalues and eigenvectors of a matrix Determination of eigenvalues and eige
Math 250-CMATLAB Assignment #61 Revised 11/11/02LAB 6: Orthonormal Bases, Orthogonal Projections, and Least SquaresIn this lab you will use MATLAB to study the following topics: Geometric aspects of vectors: the norm of a vector, the dot prod
Math 250-CMATLAB Assignment #11 Revised 9/12/01LAB 1: Matrix Computations in MATLABIn this lab you will learn how to use MATLAB to create and operate on matrices and vectors. The commands needed to do this are short and easy to remember, becau
Math 250-CMATLAB Assignment #31 Revised 12/12/01LAB 3: DeterminantsIn this lab you will use MATLAB to study the key aspects of the determinant of a square matrix: how it changes under row operations and matrix multiplication how it can be ca
Math 250-CMATLAB Assignment #41 Revised 12/11/01LAB 4: Vector Spaces and General Solution to Ax = bIn this lab you will use MATLAB to study the following topics: The subspace spanned by a set of vectors, and the fundamental concepts of indepe
Math 250-CMATLAB Assignment #51 Revised 12/12/01LAB 5: Orthogonal Subspaces, QR Factorization, and Inconsistent Linear SystemsIn this lab you will use MATLAB to study four topics: The Gram-Schmidt Algorithm that transforms a given set of basi
Math 250-CMATLAB Assignment #61 Revised 12/12/01LAB 6: Eigenvalues and EigenvectorsIn this lab you will use MATLAB to study three topics: The geometric meaning of eigenvalues and eigenvectors of a matrix Determination of eigenvalues and eige |
This edition retains all the features of the second edition, including a handy thesaurus, and adds to them by way of:
approximately 1000 new entries, primarily drawn from key areas of secondary study - technology, science, media, popular culture
additional 'Common Error' and 'Origin' b...
The Maths Tracks New South Wales Homework Books are designed to support the content of class activities and lessons. Each book consists of 40 homework pages, each with a student self-assessment task, a Homework Record and space for parent-teacher comments. A removable answer section is also provided |
Oxford International Maths for Cambridge Secondary 1
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calculus. Whether in solid mechanics or in fluid mechanics, engineers
encounter the concepts of gradients, divergence, and curl. Our
personal experience has been that engineering students are often
exceptionally talented at manipulating and computing with these vector
operations, but are much less comfortable with the relationship
between the symbols and the geometry of fluid flow. As a result when
students encounter that advanced theorems of Green, Gauss, and Stokes,
the theorems have no meaning, other than a set of symbols to memorize.
A module on the geometry of fluid flow consists of five labs in which
the students investigate the geometric meaning of gradients, divergence,
curl, flux, and path integrals. For example, in the
lab on divergence,
the students are lead to the realization that the divergence at a sink
is negative and the divergence at a source is positive. They explore
the geometry of fluid flux across a boundary and discover for
themselves that the flux across the boundary of a region is equal to
the divergence over the region (this is known as the divergence
theorem.) They also collect data that leads them to conjecture that
the divergence at a point is a limit of the flux across the boundary
of a ball centered at the point of interest as the radius of that ball
shrinks to zero.
A subsequent lab introduces students to the
geometry of curl, and
shows them movies of model fluid flows (see Figure 2). By
computing the curl at various locations and comparing their answers to
the behavior of the fluid near those points in the movies, students
develop an intuition for the geometry of curl. Essentially, the main
idea is to imagine yourself on a raft floating down a river. You drop
a tennis ball to the left of the raft and another to your right. If
the tennis balls move at different speeds than your raft, then the
river has curl. The analogous situation for three dimensional fluid
flow in more complicated, but also accessible. |
This is a full
instructional system for a first course in Algebra. It is designed for students
with an adequate arithmetic proficiency. The instruction develops a base for the
further study of Algebra. It begins by explaining basic algebraic concepts with
whole number examples. As the concepts are developed, the examples are expanded
to signed numbers (integers) and then fractions (rational numbers).
Every objective
is thoroughly explained and developed. Numerous examples illustrate concepts and
procedures. Students are encouraged to work through partial examples. Each unit
ends with an exercise specifically designed to evaluate the extent to which the
objectives have been learned. The student is always informed of any skills that
were not mastered.
The instruction
depends only upon reasonable reading skills and conscientious study habits. With
those skills and attitudes, the student is assured a successful math learning
experience. |
The study of the symmetric groups forms one of the basic building blocks of modern group theory. This book is the first completely detailed and self-contained presentation of the wealth of information now known on the projective representations of the symmetric and alternating groups.Prerequisites are a basic familiarity with the elementary theory … |
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This workbook was written to support a short course that provides a review of algebra topics designed for students who have had these subjects before. The objective is to save the student's time and money by placing them in the appropriate level math course – neither too high nor too low. If we are successful in doing this, we will provide a better learning environment for students and instructors alike.
The book has 38 separate lessons and two practice tests. Each lesson has an instructional area followed by some exercises. After the exercises there is an answer section that shows how each of the exercises is done. About midway through the book, there is a practice elementary algebra test. Then, near the end of the book, there is a practice Intermediate Algebra test. These tests are similar to the tests you will be given during the course. At the end of the book, there are extra problem sets for each lesson followed by the answers to those problem sets. |
This demanding course is designed to prepare students for college-level and AP (Advanced Placement) mathematics courses. This course represents an extension of Algebra 1 and introduces advanced algebraic skills and concepts. Major units of study include: Review of Algebra 1, Graphs and Functions, Systems and Matrices, Quadratic Equations and Functions, Polynomial Expressions and Functions, Rational Expressions and Functions, Exponential and Logarithmic Functions, Sequences and Series, an Introduction to Trigonometry, and additional topics as time permits. Students should provide their own graphing calculator (approx. $100 retail or contact the Beacon). |
The concept of function is central
to undergraduate mathematics,
foundational to modern mathematics,
and essential in related areas of
the sciences. A strong understanding
of the function concept is also
essential for any student hoping to
understand calculus - a critical
course for the development of future
scientists, engineers, and mathematicians.
This article provides an overview of what
is involved in knowing and learning the
function concept. We have included
discussions of the reasoning abilities
involved in understanding and using
functions, including the dynamic
conceptualizations needed for
understanding major concepts of calculus.
Our discussion also provides information
about the common conceptual obstacles that
have been observed in students. We make
frequent use of examples to illustrate the
'ways of thinking' and major understandings
that research suggests are essential for
students' effective use of functions during
problem solving; and that are needed for
students' continued mathematics learning.
It is noteworthy that many of the reform
calculus texts of the early 90's, e.g.,
Ostabee-Zorn (1997),
Harvard Calculus (Hughes-Hallett
& Gleason, 1994), and C4L (
Dubinsky, Schwingendorf, & Mathews, 1994),
included a stronger conceptual orientation to
learning functions. Such past curriculum
development projects and the educational
research literature are pointing the way
for future curricular interventions to
assist students in developing a robust function
conception - a conception that includes a view
of function as an entity that accepts input and
produces output and enables reasoning about
dynamic mathematical content and scientific
contexts. Research suggests that the
predominant approach to calculus instruction
is not achieving the foundational understandings
and problem solving behaviors that are needed
for students' continued mathematical development
and course taking. It is our view that the
mathematics community is ready for a careful
rethinking of the precalculus and calculus
curriculum - one that is driven by past work of
mathematicians, as well as the broad body of
research on knowing and learning function and
major concepts of calculus. It is also our
view that if algebraic and procedural methods
were more connected to conceptual learning,
students would be better equipped to apply
their algebraic techniques appropriately in
solving novel problems and tasks.
It is also common for developing students to
have difficulty distinguishing between an
algebraically defined function and an equation
(Carlson, 1998). This is
not surprising if one considers the various
uses of the equal sign and the fact that many
instructors refer to a formula as an equation.
For the student, this ambiguous use of the
word equation appears to cause difficulty for
them in distinguishing between the use of the
equal sign as a means of defining a
relationship between two varying quantities,
and a statement of equality of two expressions.
Our recent work has shown that students benefit
from explicit effort to help them distinguish
between functions and equations. We have
developed instructional interventions that
promote students' thinking about an equation
as a means of equating the output values of
two functions, and the act of solving an
equation as finding the input value(s) where
the output values of these functions are equal.
Many students also tend to believe
that all functions should be definable
by a single algebraic formula. This
focus often hinders flexible thinking
about function situations and can
lead to erroneous conclusions such as
thinking that all functions must
always behave "nicely" in some sense
(Breidenbach et al., 1992).
For example, many students tend to argue
that a piecewise defined function
like
is actually two separate functions or
that a function such as Dirichlet's
example,
is not even a function at all because it
"behaves badly." Similarly, many
students have difficulty conceiving of
different formulas representing the
same function, as in the examples
and
which define the same function on the
natural numbers, albeit through very
different algebraic operations. Many
students also tend to assume that
functions are linear or quadratic in
cases where this assumption is
unwarranted, expecting for example,
that any "u-shaped" graph is a
parabola (Schwarz &
Hershkowitz, 1999). These
confusions are perhaps not so surprising
since functions are typically introduced
in the school curriculum through specific
function types, often linear or
quadratic. Under such circumstances, a
working definition in which functions
are equated with formulas may seem
perfectly reasonable to students, and
even mirrors the historical
understanding of mathematicians like
Euler, Bernoulli, Lagrange, and
d'Alembert (Kleiner,
1989; Sierpinska,
1992). It is not, however, the view
that the Euler himself, and subsequently
the mathematics community in general,
ultimately found to be most useful. We
recommend that early function curriculum
and instruction include more opportunities
for students to experience diverse
function types emphasizing multiple
representations of the same functions.
This would promote a more flexible and
robust view of functions - one that
does not lead to inadvertently
equating functions and formulas.
Another common difficulty for students
is distinguishing between visual
attributes of a physical situation and
similar attributes of the graph of a
function that models the situation.
Throughout the secondary school
curriculum, we teach students to
attend to increasingly subtle
features of graphical representations
of functions, ranging from extrema
and monotonicity to concavity,
inflection points, and curvature.
When dealing with functions modeling
concrete situations, there are often
similar topographical structures within
the real-world setting itself (e.g.,
the curves of a racetrack, the
elevation of a road traveling across
hilly terrain, or the shape of a
container being filled with liquid).
The considerable salience of these
physical features often creates
confusion, even for students with a
strong understanding of function.
Several types of errors can be traced
to conflating the shape of a graph
with visual attributes of the
situation (Carlson,
1998; Monk,
1992; Monk &
Nemirovsky, 1994). Consider the
following problem:
The following diagram is the side-view
of an individual cycling up and over
a hill. Draw a graph of speed vs.
position along the path.
Figure 1. Students often
confuse physical features of the context
with the graph of a function.
In response to this problem, many students
tend to directly copy features of the
diagram into their graph (
Monk, 1992). Correct interpretation of the
situation is a conceptually nontrivial
task. A student must ignore the fact that
the picture looks like a graph, think of
how riding uphill (for example) affects the
speed of the cyclist, then while ignoring
the shape of the hill in the picture
determine how to represent the result
graphically.
When interpreting graphs such as the
ones in Figure 2, students often
confuse velocity for position (Monk, 1992) since the
curves are laid out spatially, and
position refers to a spatial
property. This confusion leads to
erroneous claims such as: the two
cars collide at t = 1 hour or
that Car B is catching up to Car A
between t = .75 hour and
t = 1 hour. In one study, 88%
of students who had earned a high-A
in college algebra made such
mistakes, as did 63% of students
earning an A in second semester
calculus, and 42% of students
earning an A in their first graduate
mathematics course (Carlson, 1998).
Figure 2. Students often confuse position
and velocity.
In both these examples, students are
thinking of the graph of a function as a
picture of a physical situation rather
than as a mapping from a set of input
values to a set of output values.
Developing an understanding of function
in such real-world situations that model
dynamic change is an important bridge for
success in advanced mathematics.
Students' procedural orientations to
functions have also been observed in
their inability to accurately
express function relationships using
function notation. When asked to
express s as a function of
t, many high performing
precalculus students did not know
that their objective was to write a
formula in the form of "s =
[some expression containing a t]."
Some students have also exhibited
weaknesses in knowing what each
symbol in an algebraically defined
function means. Even in the case of
a simple function such as f
(x) = 3x many students
are unaware that the parentheses
serves as a marker for the input,
that f (x) represents
the output values, that f is
the name of the function, and that
3x specifies how the input
x is mapped to the output
f (x). Such weak
understandings and highly procedural
orientations are often displayed in
the inability to move fluidly
between various function
representations, such as the
inability to construct a formula
given a function situation
described in words (Carlson, 1998).
Understanding limits and continuity
requires one to make judgments about the
value of a function on intervals of
infinitely many different (small) sizes.
Conceptualizations based on "holes,"
"poles," and "jumps" as gestalt
topographical features (corresponding to
removable discontinuity, vertical
asymptotes, and jump or one-sided
discontinuity, respectively) can lead to
misconceptions in more complex limiting
situations, such as the definitions of
the derivative and definite integral.
For example, students often develop some
intuitive understanding of the
Fundamental Theorem of Calculus with
which they can explain that the
derivative of the volume of a sphere,
is its surface area; however, most of
these students cannot explain why the same
is not true for the volume of a cube,
v = s3
(Oehrtman, 2002).
In order to resolve such results
conceptually, one must be able to
coordinate mental images of changes in
the "radius" with the corresponding
changes in the volume over a range of
small variations. For such variations,
students must then be able to imagine the
computation of rate of change of volume
and see its connection to surface area.
To understand the relationship between
average and instantaneous rates and
the graphical analog between secant
and tangent lines, a student must
first conceive of an image as in
Figure 3a, below (Monk,
1987). By employing
covariational reasoning (e.g.,
coordinating an image of two varying
quantities and attending to how they
change in relation to each other), the
student is able to transform the image
and reason about values of various
parameters as the configuration
changes. Being able to answer
questions that require such variation
as "When point Q moves toward
P, does the slope of S
increase or decrease?" is
significantly more difficult than
being able to answer questions about
the value of a function at a single
point.
Figure 3. Foundational images
for the definitions of a) the
derivative and b) the definite
integral
Analyzing the changing nature of an
instantaneous rate also requires
the ability to conceive of
functional situations dynamically.
Consider the following question
based on a classic related rates
problem in calculus:
From a vertical position against
a wall, the bottom of a ladder
is pulled away at a constant
rate. Describe the speed of the
top of the ladder as it slides
down the wall.
Reasoning about this situation
conceptually is difficult for
calculus students even when
they are given a physical model
and scaffolding questions (Monk, 1992)
and is similarly challenging for
beginning graduate students in
mathematics (Carlson, 1999).
The standard calculus curriculum
presents accumulation in terms
of methods of determining static
quantities such as the area of
an irregular region of the plane
or the total distance traveled
given a changing velocity (but
as a completed motion). Equally
important, however, is a dynamic
view in which an accumulated total
is changing through continual
accruals (Kaput,
1994; Thompson,
1994). For example, in a
typical "area so far" function as
in Figure 3b, this involves being
able to mentally imagine the point
p moving to the right by
adding slices of area at a rate
proportional to the height of the
graph. This requires students to
engage in covariational reasoning
(Carlson, Smith, &
Persson, 2003) and is
significantly more difficult for
students than evaluating and even
comparing areas at given points
(Monk, 1987).
In interviews with over 40
precalculus level students, we
found that students who
consistently verbalized a view
of function as an entity that
accepts input and produces
output were able to reason
effectively through a variety
of function-related tasks. For
example, these students, when
asked to find the composition of
f with g, fg,
given either a table or words
that defined the functions f
and g, described a process
of inputting a value into f,
with the output of f
becoming the input of g, and
this output providing an output
for the composite function, fg. However, students
who provided an incorrect answer
to this question were typically
attempting to employ some memorized
procedure. Without understanding,
they invariably made a crucial
mistake along the way such as
interpreting f (g(3)) as
meaning "the value of f when
g is three," and by mistaking
the output of g to be
3, arriving at f (3) as an
answer. As another example, when
asked to solve the equation,
f (x) = 7, given the
graph of f, students who
viewed this problem as a request to
reverse the function process to
determine the input associated with
an output of f, had no
difficulty responding to this task.
Surprisingly, only 38% of 1196
students (550 college algebra and
646 precalculus) provided a correct
answer at the completion of their
courses. Those unable to provide a
correct answer appeared to be
applying memorized procedures - they
did not speak about a function as a
more general mapping of a set of
input values to a set of output
values. Their impoverished function
view was also revealed by their
inability to explain the meaning of
function composition and function
inverse in other settings and their
inability to apply function
composition to define an algebraic
formula for a function situation
(e.g., to define area as a function
of time for a circle whose radius is
expanding at 7 cm per second).
An action conception of
function would involve the ability to
plug numbers into an algebraic
expression and calculate. It is a
static conception in that the subject
will tend to think about it one step
at a time (e.g., one evaluation of an
expression). A student whose function
conception is limited to actions might
be able to form the composition of two
functions, defined by algebraic
expressions, by replacing each
occurrence of the variable in one
expression by the other expression and
then simplifying; however, the
students would probably be unable to
compose two functions that are defined
by tables or graphs.
Students whose understanding is limited
to an action view of function
experience several difficulties. For
example, an inability to interpret
functions more broadly than by the
computations involved in a specific
formula results in misconceptions such
as believing that a piecewise function
is actually several distinct functions,
or that different algorithms must produce
different functions. More importantly,
reasoning dynamically is difficult
because it requires one to be able to
disregard specific computations and to
be able to imagine running through
several input-output pairs simultaneously.
This ability is not possible with an
action view in which each individual
computation must be explicitly performed
or imagined. Furthermore, from an action
view, input and output are not conceived
except as a result of values considered
one at a time, so the student cannot
reason about a function acting on entire
intervals. Thus, not only is the
complex reasoning required for calculus
out of reach for these students, but
even simple tasks like conceiving of
domain and range as entire sets of
inputs and outputs is difficult.
Without a generalized view of inputs
and outputs, students cannot think of
a function as a process that may be
reversed (to obtain the inverse of a
function) but are limited to
understanding only the related
procedural tasks such as switching
x and y and solving for
y or reflecting the graph of
f across the line y
= x (Figures 4a and 4b).
This procedural approach to determining
"an answer" has little or no real
meaning for the student unless he or
she also possesses an understanding as
to why the procedure works. Students
with an action view often think of a
function's graph as being only a curve
(or fixed object) in the plane; they
do not view the graph as defining a
general mapping of a set of input
values to a set of output values. As
such, the location of points, the
vertical line test, and the "up and
over" evaluation of functions on a
graph are concepts only about the
geometry of the graph, not about the
more general mapping that is conveyed
by the function, or the meaning that is
conveyed by inverting the process for a
function that represents a real-world
situation. Similarly, with an action
view, composition is generally seen
simply as an algebra problem in which
the task is to substitute one expression
for every instance of x into
some other expression. An understanding
of why these procedures work or how they
are related to composing or reversing
functions is generally absent.
Figure 4. Various conceptions of
the inverse of a function. a) as an
algebra problem, b) as a geometry
problem, and c) as the reversal of a
process. The first two of these are
common among students but, in
isolation, do not facilitate flexible
and powerful reasoning about
functional situations.
Students who possess only the
procedural orientations of
Figures 4a and b, absent of
the understanding of why the
procedures work, will likely be
unable to recognize even simple
situations in which these
procedures should be applied.
Curriculum and instruction have
not been broadly effective in
building these connections in
students' understanding. A
recent study of over 2000
precalculus students at the end
of the semester (Carlson,
Oehrtman, & Engelke,
submitted) showed that only
17% of these students were able
to determine the inverse of a
function for a specific value
given a small table of function
values.
A process conception of
function involves a dynamic
transformation of quantities
according to some repeatable
means that, given the same
original quantity, will always
produce the same transformed
quantity. The subject is able to
think about the transformation
as a complete activity beginning
with objects of some kind, doing
something to these objects, and
obtaining new objects as a result
of what was done. When the
subject has a process conception,
he or she will be able, for
example, to combine it with other
processes, or even reverse it.
Notions such as 1-1
or onto become more accessible as
the students' process conception
strengthens.
With such a process view, students
are freed from having to imagine
each individual operation for an
algebraically defined function.
For example, given the function on
the real numbers defined by
f (x) =
x2+1, the student can
imagine a set of input values that
are mapped to a set of output
values by the defining expression for
f. In contrast, students with
an action view see the defining
formula as a procedure for finding an
answer for a specific value of
x; they view the formula as a
set of directions: square the value
for x then add one to get the
answer. A student with a process
view can conceive of the entire
process as happening to all values
at once, and is able to conceptually
run through a continuum of input
values while attending to the
resulting impact on output values.
This is precisely the ability
required for covariational reasoning
introduced above and discussed more
fully in the following section. In
Table 1, we provide a
characterization of "action views"
of functions and their corresponding
"process views."
Table 1. Action and process
views of functions
Action View
Process View
A function is tied to a specific
rule, formula, or computation and
requires the completion of specific
computations and/or steps.
A function is a generalized
input-output process that defines a
mapping of a set of input values to a
set of output values.
A student must perform or imagine
each action.
A student can imagine the entire
process without having to perform
each action.
The "answer" depends on the formula.
The process is independent of the
formula.
A student can only imagine a single value
at a time as input or output (e.g.,
x stands for a specific number).
A student can imagine all input at once or
"run through" a continuum of inputs. A
function is a transformation of entire
spaces.
Composition is substituting a formula or
expression for x.
Composition is a coordination of
two input-output processes; input is
processed by one function and its output is
processed by a second function.
Inverse is about algebra (switch y
and x then solve) or geometry (reflect
across y = x).
Inverse is the reversal of a process
that defines a mapping from a set of output
values to a set of input values.
Domain and range are conceived at most as an
algebra problem (e.g., the denominator cannot
be zero, and the radicand cannot be negative).
Domain and range are produced by operating
and reflecting on the set of all possible
inputs and outputs.
Functions are conceived as static.
Functions are conceived as dynamic.
A function's graph is a geometric figure
A function's graph defines a specific
mapping of a set of input values to a set
of output values.
Understanding even the basic idea
of equality of two functions
requires a generalization of
the input-output process, the
ability to imagine the pairing
of inputs to unique outputs
without having to perform or even
consider the means by which this
is done. Students may then come
to understand that any means of
defining the same relation is the
same function. That is, a function
is not tied to specific
computations or rules that define
how to determine the output from
a given input. For example, the
rules
vs.
both provide the same results on the
natural numbers and thus define the
same function.
Students with a process view
are also better able to
understand aspects of
functions such as composition
and inverses. They are
consistently able to
correctly answer conceptual
and computational questions
about composition in a variety
of representations by
coordinating output of one
process as the input for a
second process. Similarly,
students conceiving of inverses
as reversing a process so that
the old outputs become the new
inputs and vice-versa (Figure
4c), or by asking "What does one
have to do to get back to the
original values?" were able to
correctly answer a wide variety
of questions about inverse
functions (Carlson
et al., submitted).
A process view of function is
crucial to understanding the
main conceptual strands of
calculus (Breidenbach
et al., 1992;
Monk, 1987;
Thompson, 1994a
). For example, the ability
to coordinate function inputs
and outputs dynamically is an
essential reasoning ability for
limits, derivatives, and definite
integrals. In order to
understand the definition of a
limit, a student must coordinate
an entire interval of output
values, imagine reversing the
function process, and determine
the corresponding region of
input values. The action of a
function on these values must be
considered simultaneously since
another process (one of reducing
the size of the neighborhood in
the range) must be applied while
coordinating the results.
Unfortunately, most pre-calculus
students do not develop beyond
an action view, and even strong
calculus students have a poorly
developed process view that
often leads only to
computational proficiency (Carlson, 1998).
With intentional instruction,
however, students can develop a
more robust process view of
function (Carlson et
al., submitted;
Dubinsky, 1991;
Sfard, 1991).
Certainly not every aspect of an
action view of functions is
detrimental to students'
understanding, just as the
acquisition of a process view
does not ensure success with all
functional reasoning. However, a
process view of functions is
crucial to developing rich
conceptual understandings of the
content in an introductory
calculus course. The promotion of
the more general 'ways of thinking'
that we have advocated should
result in producing curricula that
are more effective for promoting
conceptual structures for students'
continued mathematical
development.
Working out both the correct diagram and
the correct formula for the inverse
encourages students to think in terms of
a general input-output process. As
another example, students typically
learn to carry out rote procedures when
asked to solve equations such as
f (x) = 6 for
some defined function f; but
asking them to find the input value(s)
for which the function's output is 6
(both algebraically and graphically)
promotes an understanding that solving
an equation can be seen as the reversal
of a function process. As yet another
example, students typically memorize
(without understanding) that the graph of
a function g given by
g(x) = f
(x+a) is shifted
to the left of the graph of f, but
asking them to discover or interpret this
statement as meaning "the output of
g at every x is the same as
the output of f at every x+a" will give them a
more powerful way to understand this idea
and reinforce a process view. Ask
students to determine the domain and range
of functions based on the problem context,
and relate this to answers (possibly
different) derived from algebraic
constraints alone. Other possibilities
include asking students to explain why
composition is associative, to develop the
definition of a periodic function on their
own, or to graph and explain the results
of simple function arithmetic.
Ask about the action of functions on
entire intervals in addition to single
points. Focusing on the image of a
function applied to an infinite set also
encourages students to think in terms of
a general process. Students should be
asked to coordinate such judgments with
basic compositions and inverses, asking,
for example, for the length of an
interval after being transformed by two
linear functions. Similarly, ask
students to find preimages of intervals
as in the definition of limit or
continuity and to reverse the process of
a function even if it is not invertible
(e.g., find the preimages of 1 under
f (x) = x2).
Ask students to make and compare
judgments about functions across
multiple representations. Such
questions should include multiple
algebraic representations to
reinforce the independence from a
formula as well as the standard
representations of graphs, tables,
and verbal descriptions. Students
should make such determinations;
then compare the results for
consistency, justifying or
discovering why they are the same.
For example, asking how the various
techniques of inverting a function
are related reinforces seeing a
reflection across the line x = y as
switching the roles of independent
and dependent variable, of input
and output. Also helpful are
predictions about how a graph will
look based on how a real-world
quantity is changing across its
domain, requiring simultaneous
attention to multiple input-output
pairs and translation between
representations.
Labeling the axes with verbal
indications of coordinating the two
variables (e.g., y changes with
changes in x)
Mental Action 2
(MA2)
Coordinating the direction of change
of one variable with changes in the
other variable
Constructing a monotonic straight line
Verbalizing an awareness of the
direction of change of the output while
considering changes in the input
Mental Action 3
(MA3)
Coordinating the amount of change of
one variable with changes in the other
variable
Plotting points/constructing secant
lines
Verbalizing an awareness of the
amount of change of the output while
considering changes in the input
Mental Action 4
(MA4)
Coordinating the average rate-of-change
of the function with uniform increments of
change in the input variable
Constructing secant lines for
contiguous intervals in the domain
Verbalizing an awareness of the rate
of change of the output (with respect to the
input) while considering uniform increments
of the input
Mental Action 5
(MA5)
Coordinating the instantaneous
rate-of-change of the function with
continuous changes in the independent
variable for the entire domain of the
function
Constructing a smooth curve with
clear indications of concavity changes
Verbalizing an awareness of the
instantaneous changes in the
rate-of-change for the entire domain of
the function (direction of concavities
and inflection points are correct)
In our work to study and promote
students' emerging covariational
reasoning abilities, we have found
that the ability to move flexibly
between mental actions 3, 4 and 5
is not trivial for students. We
have also observed that many
precalculus level students only
employ Mental Action 1 and Mental
Action 2 when asked to construct
the graph of a dynamic function
situation.
When prompting students to construct
the graph of the height as a
function of the amount of water in a
bottle (Figure 6), we found that
many precalculus students
appropriately labeled the axes (MA1)
and then constructed an increasing
straight line (MA2). When prompted
to explain their reasoning, they
frequently indicated that "as more
water is put into the bottle, the
height of the water rises (MA2)."
These students were clearly not
attending to the amount of change
of the height of the water level or
the rate at which the water was
rising.
Imagine this bottle filling with water.
Sketch a graph of the height as a function of the amount of
water that's in the bottle.
Figure 6. The Bottle Problem.
We have observed that calculus
students frequently provided a
strictly concave up graph in
response to this question
(Carlson, 1998
; Carlson et al.,
2002). When probed to explain
their reasoning, a common type of
justification was, "as the water
is poured in it gets higher and
higher on the bottle (MA2)." In
contrast, other students who were
starting to be able to construct
an appropriate graph began
coordinating the magnitude of
changes in the height with changes
in the volume (MA3). This is
exemplified in the strategy of
imagining pouring in one cup of
water at a time and coordinating
the resulting change in height
based on how "spread out" that
layer of water is.
Other students have demonstrated
the ability to speak about the
average rate of change locally for
a specific interval of a function's
domain (MA4) but were unable to
explain how the rate changes over
the domain of the function. Even
when calculus students produced a
graph that was correct, they
commonly had difficulty explaining
what was conveyed by the
inflection point and why the graph
was "smooth" (i.e., C1 rather than
piecewise linear). Students
frequently exhibited behaviors that
gave the appearance of engaging in
Mental Action 5 (e.g.,
construction of a smooth curve with
the correct shape), however when
prompted to explain their
reasoning, they explained that
they had relied on memorized facts
to guide their constructions.
They were relying on facts such as
faster means steeper and slower
means less steep, but they were
unable to explain why this was
true. |
Those happen to coincide with some of the NCTM (National Council of Teachers of Mathematics) ``standards'' for mathematics education. We have:
The students shall ...
...develop an appreciation of mathematics, its history and its applications.
...become confident in their own ability to do mathematics.
...become mathematical problem solvers.
...learn to communicate mathematical content.
...learn to reason mathematically.
General Education Course Objectives:
Thinking Skills: Students will ...
(a)
...explore the problems that lead to differential equations;
(b)
...learn to solve basic types of first order differential equations that can be done in closed form. Those include: separation of variables, Clairaut equations, homogeneous equations, equations with integrating factors.
(c)
...learn to solve first order linear differential equation, and second order linear differential equation with constant coefficients.
(d)
...Learn geometric interpretation of first order equations, and learn constructing the corresponding vector field.
...study the use of Laplace transform in solving differential equations.
(i)
...explore solving of systems of first order differential equations.
Communication Skills: Students will ...
...justify their reasoning, provide precise, rigorous explanations for their work.
... learn how to typeset the solutions to their homework; learn to use LATEX
...learn to use the Internet resources and present the findings in class.
...learn to model a real life problem.
...learn how to interpret a solution of an ODE and apply it in a real life problem.
...learn how to prove some theorems explore the importance of differential equations in sciences and various applications, as well as become familiar with some elements of the historical development of the fieldThis course is aimed at the Math and Science majors with the idea of introducing to them some standard, most basic, topics and techniques in the field of differential equations.
Course Philosophy and Procedure
Two key components of a success in the course are regular attendance and a fair amount of constant, every-day study. You should try to make sure that your total study time per week at least triples the time spent in class.
This course will introduce you to one of the key tools of applied mathematics. In addition to applications of differential equations in solving a variety of problems, you will be able to see some of the mathematical subtleties involved, and thus further develop your mathematical maturity. Also, I hope that through exploring various advanced techniques in solving differential equations, you will see some of the ways how attempts to solve difficult differential equations problems generate development of new mathematics.
Above all, this course provides an excellent opportunity to master the basic tools of Calculus, that is, differentiation and integration.
Grading will be based on two in-class exams ( points each), a cumulative final exam ( points), class participation, take-home problems, projects, group practice exams and portfolios.
My grading scale is
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%. |
Alevel Core 1
This document contains notes for numerical solutions of equations. It contains just a few quick key points that you should remember especially for FP1 exams. Equations of the form f(x) = 0 can be solved using interval bisection. If you [...]
This section contains useful notes for differentiation. The gradient of a curve y=f(x) at a specific point is equal to the gradient of the tangent to the curve at that point. The gradient of the tangent at any particular point [...]
This chapter explores dividing polynomials. It covers algebraic division of polynomials by (x + a) or (x – a). Before attempting this chapter you must have prior knowledge of long division and knowing that polynomials can be written as a [...]
This chapter explores equation of a line. It covers understanding that y – y = m(x = x1) is the equation of a straight line, finding the equation of a line using gradient and one point, finding the equation of [...]
This chapter explores solving equations with algebraic fractions. It covers understanding how to solve equations involving fractions, working with denominators with either constants or linear factors. Before attempting this chapter you must have prior knowledge of expanding brackets and factorising [...]
This section explores intersecting lines and curves. This is the first part of this chapter. It covers finding the intersection of a line and a curve using algebraic methods. It will be useful to have prior knowledge of solving linear [...]
This chapter explores sketching polynomials. It covers general shapes of quadratics, cubics and identifying the maximum number of turning points, roots and intercepts, and linking to factorised forms for example f(x) = (x-a)(x-b)(x-c). Before attempting this chapter you must have [...]
This chapter explores particular solutions to differential equations. The chapter covers finding the particular solution to a first order differential equations by suing given conditions. Before attempting this chapter you must have prior knowledge of solving first order differential equations [...] |
Introductory Mathematical Analysis by W.P. Webber, L.C. Plant - John Wiley & sons , 1919 The present text is the result of several years of study and trial in the classroom in an effort to make an introduction to college mathematics more effective and better suited to its place in a scheme of education under modern conditions of life. ( 280 views) Calculus in Context by James Callahan, et al. - Five Colleges, Inc. , 2008 In this course you will learn to use calculus both as a tool and as a language in which you can think coherently about the problems you will be studying.
Exam Content In 1956, 386 students took what was then known as the AP Mathematics Exam. By 1969, still under the heading of AP Mathematics, it had become Calculus AB and Calculus BC. The Calculus BC exam
You'll find new and improved student resources in Explore AP , the new AP site for students The AP Calculus BC Exam |
Chapter 7: Sequences and Series
Section 1 Overview of Sequences and Series
Section 2 Arithmetic Sequences and Series
Section 3 Geometric Sequences and Series
Section 4 Mathematical Induction and the Binomial Theorem |
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Helpful Links
Self-Help
Cambridge's University Counselling Services provides a broad array of leaflets ranging from bolstering study skills and preparing for exams to advice on coping with homesickness or loneliness: link.
Math Anxiety
Professor Diane Johnson from Humboldt State University, CA has allowed me to share a literature review she completed on the topic of math anxiety: what it is, causes, effects, and treatment. It is definitely worth reading by students and teachers alike. Here is the link.
Study Skills and Exam Preparedness
There are plenty of web sites that address these topics. Other than the leaflets provided by the link at the top of this page, Paul Dawkins from Lamar University has an excellent site with advice on how to study mathematics, take notes, prepare for exams, etc. This is essential reading for students:link. You will find Imperial College and Reach Out also offer good advice for study habits and ways to prepare for exams. Finally, for a touch of humor, Martin Greenhow of Brunel University offers these tidbits: link.
Homework
It is assumed, incorrectly, that students at the college level have a good grasp of homework guidelines for mathematics. For those students needing a refresher, Purplemath offers homework guidelines to help you better communicate with your instructor. This is essential reading for students:link.
Algebra Help
Purplemath to the rescue! This link has further lessons and examples of common algebra topics. SOS Math: Algebra also has many good explanations, examples, and drills/practice problems
More Examples/Explanations
SOS Math is a fantastic resource for students who need help with the subjects of Algebra, Trigonometry, Calculus, Differential Equations, Complex Variables, and Matrix Algebra.
Graphing Calculator Assistance
Prentice Hall has a site with step-by-step directions on how to use the features of most Texas Instruments graphing calculators (for those with TI-84's you can follow the instructions for the TI-83) and a few non-TI graphing calculators like the Casio FX2 and Hewlett-Packard 48G, for example: link.
Wolfram Mathematica Online Integrator
Here is something for those folks wanting to check if they are on the right track with their integration: link. Keep in mind that Wolfram does not include a constant of integration for indefinite integrals.
Wolfram Alpha Computational Knowledge Engine
Wolfram offers a really slick web resource with its search engine extraordinaire. Here is a link to the mathematics examples page: link. This is an excellent resource for students.
Common Math Errors
Please read the web page entitled, "The Most Common Errors in Undergraduate Mathematics." Here is the link. After reading the article, please avoid falling into these traps in your current and future math courses. Good luck! |
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LibreOffice - Math (LibreOffice Math)
Math is the LibreOffice suite's formula editor, that can be invoked in your text documents, spreadsheets, presentations and drawings, to enable you to insert perfectly-formatted mathematical and scientific formulas. Your formulas can include a wide range of elements, from fractions, terms with exponents and indices, integrals, and mathematical functions, to inequalities, systems of equations, and matrices.
More info »
Links
You're browsing LibreOffice - MathLibreOffice - MathLatexFormula is an easy-to-use graphical application for generating images (that you can drag and drop, copy and paste or save to disk) from LaTeX equations. It is written in Qt4. These images can be...
Math, part of the Apache OpenOffice suite, is a tool for creating and editing mathematical equations, similar to Microsoft Equation Editor (included in Microsoft Office). The created equations can then be...
MathCast is an equation editor, an application that allows you to input mathematical equations. These equations can be used in written documents, webpages, and even databases. They could be rendered...
Equation Illustrator V home page. Programs for the production of single page hand out type documents of a technical nature. Documents can include furmulea,vector graphic drawings and pictures. Enhanced |
Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate.
Reviews
P.R. Halmos Measure Theory "As with the first edition, this considerably improved volume will serve the interested student to find his way to active and creative work in the field of Hilbert space theory."--MATHEMATICAL REVIEWS
You can earn a 5% commission by selling Measure Theory: v. 18 (Graduate Texts in Mathematics |
Friendly Introduction to Number Theory nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers.Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Pro... MOREvides a new chapter that introduces the theory of continued fractions. Includes a new chapter on "Continued Fractions, Square Roots and Pellrs"s Equation." Contains additional historical material, including material on Pellrs"s equation and the Chinese Remainder Theorem.A useful reference for mathematics teachers. For courses in Elementary Number Theory for math majors, for mathematics education students, and for Computer Science students. This introductory undergraduate text is designed to entice a wide variety of majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. the writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results. |
Course Offerings
Mathematics Courses
Math 130. Basic Math – Algebra and Trigonometry. 3(1). This course sis designed to help reinforce algebraic and trigonometric skills necessary for success in the technical core. Basic graphing, algebraic manipulation, and trigonometric calculations are covered. This course may be used as an Academy option to fulfill graduation requirements. This course does not fulfill any major's requirements. Final Exam. Prereq: Can only be enrolled in the course by recommendation of the Department of Mathematical Sciences. Sem hrs: 3 fall.
Math 141.Calculus I. 3 (1). The study of differential calculus. Topics include functions and their applications to physical systems; limits and continuity; vectors and vector arithmetic; a formal treatment of derivatives; numeric estimation of derivatives at a point; basic differentiation formulas for elementary functions; product, quotient, and chain rules; implicit differentiation; and mathematical and physical applications of the derivative, to include extrema, concavity, and optimization. Significant emphasis is placed on using technology to solve and investigate mathematical problems. Final exam. Sem hrs: 3 fall or spring.
Math 142. Calculus II. 3(1). A study of integral calculus with a focus on the Fundamental Theorems and their application. Topics include: estimating area under a curve, antiderivatives, numeric integration methods, antiderivative formulas for the elementary functions, integration by substitution, parts and tables, improper integrals, differential equations, exponential growth and decay, an introduction to Taylor Series, and mathematical and physical applications of the Fundamental Theorems. Physical applications include area and volume problems and the concept of work. Final exam. Prereq: Math 141. Sem hrs: 3 fall or spring.
Math 152. Advanced Placed Calculus II. 3(1). A more intense study of integral calculus for advanced-placed fourth-class cadets. Content is similar to Math 142, with the addition of an introduction to polar coordinates, vector arithmetic, and complex arithmetic. Additional emphasis is placed on the mathematical and physical applications in preparation for cadets interested in pursuing a technical major or minor. Final exam. Prereq: For fourth-class cadets--qualifying performance on DFMS placement exams; for third-class cadets, Department Head approval. Sem hrs: 3 fall.
Math 243. Calculus III. 3(1). Multivariate calculus, including vector functions, partial differentiation, directional derivatives, line integrals, and multiple integration. Maxima and minima in multiple dimensions and the method of Lagrange Multipliers. Solid analytical geometry to include lines, planes, and surfaces in 3-space. Designed for cadets who indicate an interest in a technical major. Final exam. Prereq: C or better in Math 142 or advanced-placement through DFMS exams. Waiver authority is Deputy Head for Academics. Sem hrs: 3 fall or spring.
Math 300. Introduction to Statistics. 3(1). Topics include descriptive statistics, emphasizing graphical displays; basic probability and probability distributions; sampling distribution of the mean and the Central Limit Theorem; statistical inference including confidence intervals and hypothesis testing; correlation; and regression. Math 300 is designed primarily for majors in the Social Sciences and Humanities. It emphasizes the elements of statistical thinking, focuses on concepts, automates most computations, and has less mathematical rigor than Math 356. Final exam. Prereq: Math 142/152 or department permission. Sem hrs: 3 fall or spring.
Math 310. Mathematical Modeling. 3(1). An introductory course in mathematical modeling. Students model various aspects of real-world situations chosen from Air Force applications and from across academic disciplines, including military sciences, operations research, economics, management, and life sciences. Topics include: the modeling process, graphical models, proportionality, model fitting, optimization, and dynamical systems. Several class periods are devoted to in-class work on small projects. Math 310 is not appropriate for Math or OR majors. Final exam. Prereq: Completion of core math. Sem hrs: 3 spring.
Math 320. Foundations of Mathematics. 3(1). Course emphasizes exploration, conjecture, methods of proof, ability to read, write, speak, and think in mathematical terms. Includes an introduction to the theory of sets, relations, and functions. Topics from algebra, analysis, or discrete mathematics may be introduced. A cadet cannot receive credit for both Math 320 and Math 340. Final exam or final project. Prereq: Completed Math 142/152 with a 'C' or better. Wavier authority is the Deputy Head for Academics. Sem hrs: 3 fall or spring.
Math 356. Probability and Statistics for Engineers and Scientists. 3(1). Topics include classical discrete and continuous probability distributions; generalized univariate and bivariate distributions with associated joint, conditional, and marginal distributions; expectations of random variables; Central Limit Theorem with applications in confidence intervals and hypothesis testing; regression; and analysis of variance. This course is a core substitute for Math 300. Credit will not be given for both Math 300 and Math 356, not for both Math 356 and Math 377. Designed for cadets in engineering, science, or other technical disciplines. Math majors and Operations Research majors should take the Math 377/378 sequence. Final Exam. Prereq: Math 142/152. Sem hrs: 3 fall or spring.
Math 359. Design and Analysis of Experiments. 3(1). An introduction to the philosophy of experimentation and the study of statistical designs. The course requires a knowledge of statistics at the Math 300 level. Topics include analysis of variance for K treatments, various two- and three-level designs, interactions, unbalanced designs, and regression analysis. A valuable course for all science and engineering majors. Final project. Prereq: Math 300, Math 356 or Math 378. Sem hrs: 3 spring.
Math 360. Linear Algebra. 3(1). A first course in linear algebra focusing on Euclidean vector spaces and their bases. Using matrices to represent linear transformation, and to solve systems of equations, is a central theme. Emphasizes theoretical foundations (computational aspects are covered in Math 344). A cadet cannot receive credit for both Math 344 and Math 360. Final exam or final project. Prereq/Coreq: Math 320 or department permission. Sem hrs: 3 fall.
Math 366. Real Analysis I. 3(1). A theoretical study of functions of one variable focused on proving results related to concepts first introduced in differential and integral calculus. This course is an essential prerequisite for graduate work in mathematical analysis, differential equations, optimization, and numerical analysis. Final exam or final project. Prereq: Math 360 or department permission. Sem hrs: 3 spring.
Math 370. Introduction to Point-Set Topology. 3(1). Review of set theory; topology on the real line and on the real plane; metric spaces; abstract topological spaces with emphasis on bases; connectedness and compactness. Other topics such as quotient spaces and the separation axioms may be included. A valuable course for all math majors in the graduate school option. Final exam. Prereq: Math 320. Sem hrs: 3 spring of even years.
Math 372. Introduction to Number Theory. 3(1). Basic facts about integers, the Euclidean algorithm, prime numbers, congruencies and modular arithmetic, perfect numbers and the Legendre symbol will be covered and used as tools for the proof of quadratic reciprocity. Special topics such as public key cryptography and the Riemann Zeta function will be covered as time allows. Final exam. Prereq: Math 320 Sem hrs: 3 spring of odd years.
Math 378. Advanced Statistics. 3(1). Topics include point and interval estimation, properties of point estimators, sample inferential statistics with confidence intervals, hypothesis testing, ANOVA, linear regression, design and analysis of experiments, and nonparametric statistics. This course is a core substitute for Math 300 but has much more rigor and depth. Credit will not be given for both Math 300 and Math 378. Final exam. Prereq: Math 377. Sem hrs: 3 spring.
Math 421. Mathematics Capstone II. 3(1). The second semester of the mathematics capstone experience. Students will compete work on their independent research project and produce a thesis to present their findings. Final project. Prereq: C1C standing in the Mathematics major. Sem hrs: 3 spring.
Math 443. Numerical Analysis of differential Equations. 3(1). An intermediate numerical analysis course with an emphasis on solving differential equations. Specific topics include solving linear and nonlinear systems; solutions of initial value problems via Runge-Kutta, Taylor, and multistep methods; convergence and stability; and solutions of boundary value problems. Other topics typically include approximating eigenvalues and eigenvectors and numerically solving partial differential equations. The approach is a balance between the theoretical and applied perspectives with some computer programming required. Final exam or final report. Prereq: Math 346 and one of Math 342 or Physics 356, or department permission. Sem hrs: 3 spring of even-numbered years.
Operations Research Courses
These courses are offered in one of the following departments: Department of Computer Science (DFCS), Department of Economics and Geography (DFEG), Department of Management (DFM), and Department of Mathematical Sciences (DFMS)
Ops Rsch 310. Systems Analysis. 3(1). This course provides an introduction to rigorous quantitative modeling methods that have broad application. The course focuses on the mathematics of the models, the computer implementation of the models, and the application of these models to practical decision-making scenarios. By demonstrating the application of these techniques to problems in a wide range of disciplines, the course is relevant to technical and non-technical majors at USAFA. The course consists of six distinct blocks: decision analysis and utility theory, linear and nonlinear optimization, project management, queuing theory, simulation, and the systems approach to engineering and decision-making. Administered by the Department of Management. Instruction provided by inter-departmental Operations Research faculty. Final exam. Prereq: Comp Sci 110, Math 142. Sem hrs: 3 Fall and Spring.
Ops Rsch 405. First-Class Seminar. 0(1). A course for First-Class Operations Research majors that provides for presentation of cadet and faculty research; guest lectures; field trips; seminars on career and graduate school opportunities for scientific analysts in the Air Force; goal setting exercises; and applications of Operations Research. The class meets once each week. Open only to First-Class Operations Research majors. Pass/Fail. No final exam. Prereq: C1C standing. Sem hrs: 0 fall.
Ops Rsch 411. Topics in Mathematical Programming. 3(1). Topics include linear programming (with sensitivity analysis and applications) and non-linear programming. Both the theory and the computer implementation of these techniques are addressed. Administered by the Department of Mathematical Sciences. Final exam. Prereq: Math 343 or Math 360; and either Ops Rsch 310 or department permission. Sem hrs: 3 fall.
Ops Rsch 419. Case Studies in Operations Research.1.5(1) . The study of methodologies associated with business and operations management. A case-based course intended to provide the proper foundation needed to conduct effective analyses supporting a variety of scenarios. Students will evaluate various cases, develop plans for and conduct analyses, and create effective written and oral presentations. Final Project or Exam. Sem hrs: 1.5 fall. |
For Christian & Home Schools
Logic is the art of reasoning well—Designed for 8th-grade and up, the lessons in this text cover definitions, logical statements, fallacies, syllogisms, and many other elements. This course is a thorough introduction and serves as both a self-contained course and a preparatory course for more advanced studies.
Together with Intermediate Logic, this text
Changes in Introductory Logic 4th Edition
This curriculum has been changed in a number of critical areas:
First, in order to present to the student a more logical progression of topics, the section on defining terms has been moved from Intermediate Logic to Introductory Logic, where it is taught along with other branches of informal logic and categorical logic.
Second, review questions and review exercises have been added to each unit for every lesson in the text, effectively doubling the number of exercises for students to verify their knowledge and develop their understanding of the material. Additionally, some especially challenging problems have been included in the review exercises.
Third, the definitions of important terms, key points made, and caution signs regarding common errors are now set apart in the margins of the text. This should help students to distinguish the most important topics, as well as aid in their review of the material.
Fourth, every lesson has been reviewed in great detail with the goal of improving the clarity of the explanations and correcting several minor errors that were found in the original edition.
Fifth, the book is now in a handsome perfect-binding, with all exercise pages perforated for easy removal. |
Daily Warm-Ups Algebra for Common Core State Standards
Each title in our new set of Daily Warm-Ups contains more than 100 focused activities to challenge your studentsí thinking. These three books support implementation of the Common Core State Math Standards, including the Common Core Mathematical Practices, with a firm foundation of important concepts and problem-solving skills.
Daily Warm-Ups: Algebra for Common Core State Standards features problems addressing the following topics: Number and Quantity; Algebra; Functions; and Statistics and Probability.
Materials include:
Reproducible teacher book
More than 100 varied problems directly addressing CCSS
Includes CD-ROM with detailed correlations, student problems ideal for projecting within the classroom, and an answer key |
Exploring, Investigating and Discovering in Mathematics
May 23, 2011 - 22:52 — Anonymous
Author(s):
V. Berinde
Publisher:
Birkhäuser
Year:
2003
ISBN:
3-7643-7019-X
Price (tentative):
€34
MSC main category:
00 General
Review:
The book is a collection of problems from elementary mathematics. It can be of substantial help in work with gifted secondary school students. On the other hand, it also contains problems on determinants, special sequences, functional equations, primitive functions, difference and differential equations, so that it will be useful for work with students of basic courses on analysis and algebra. The collection is divided into 24 groups. Over 100 problems are presented with solutions, and another 150 are accompanied by hints and clear ideas how to proceed on the way to a solution. Using included material, the author leads readers from active problem solving to exploration of methods to obtain new problems and to an active use of the gained inventive skills. The book is based on the author's personal long lasting cooperation with the Romanian journal Gazeta Matematica. |
Election Analytics - University of Illinois
Up-to-the-minute estimates for the probabilities of all federal elections that take place in a year when Americans vote for U.S. President: who will assume the presidency, who will control the United States Senate, and the House of Representatives. With
...more>>
Elementary Computer Mathematics - Kenneth R. Koehler
An introduction to the mathematics used in the design of computer and network hardware and software. This hypertextbook's goal is to prepare the student for further coursework in such areas as hardware architecture, operating systems internals, applicationElliptic 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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Elsevier Science
"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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The Elverson Puzzle Co., Inc.
The creators of Jacob's Revenge, the classic wine bottle puzzle. Elverson Puzzle also makes a Tangram Puzzle, the eleven tiles of which form a 6 inch diameter circle; Farkle, a game played with extra large wooden dice or online, with the dice roller that
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eMathHelp
View worked solutions to problems, or submit your own to WyzAnt.com tutors. See also eMathHelp's notes on pre-algebra, algebra, calculus, differential equations, and more.
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Emaths.Info - Vinod Sebastian
Math tools, formulas, tutorials, videos, tables, and "curios," such as the unusual properties of 153, 1729, and 2519. See in particular Emaths' interactive games, which include the N queens problem, Towers of Hanoi, and partition magic, which finds a
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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.
...more>>
The EMPower Project - TERC
Extending Mathematical Power: A Math Curriculum for Adults, extends school mathematics reform to currently under-served populations so that they more effectively engage mathematical demands at work, at home as parents and caregivers, in the community,The Endeavour - John D. Cook
Cook's blog posts, which date back to January, 2008, have included "A Ramanujan series for calculating pi," "Limerick primes," "Twin prime conjecture and the Pentium division bug," "Three surprises with the trapezoid rule," "Where to wait for an elevator,"
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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
...more>> |
Precalculus Essentials, CourseSmart eTextbook Essentials' table of contents is based on learning objectives and condensed to cover only the essential topics needed to be successful in calculus. Using the text with MyMathLab, students now have access to even more tools to help them be successful including "just-in-time" review of prerequisite topics right when they need it.
This text offers a fast pace and includes more rigorous topics ideal for students heading into calculus.
Table of Contents
P. Basic Concepts of Algebra
P.1 The Real Numbers; Integer Exponents
P.2 Radicals and Rational Exponents
P.3 Solving Equations
P.4 Inequalities
P.5 Complex Numbers
1. Graphs and Functions
1.1 Graphs of Equations
1.2 Lines
1.3 Functions
1.4 A Library of Functions
1.5 Transformations of Functions
1.6 Combining Functions; Composite Functions
1.7 Inverse Functions
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Exercises for Calculus
Chapter 1 Practice Test A
Chapter 1 Practice Test B
2. Polynomial and Rational Functions
2.1 Quadratic Functions
2.2 Polynomial Functions
2.3 Dividing Polynomials and the Rational Zeros Test
2.4 Zeros of a Polynomial Function
2.5 Rational Functions
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Exercises for Calculus
Chapter 2 Practice Test A
Chapter 2 Practice Test B
3. Exponential and Logarithmic Functions
3.1 Exponential Functions
3.2 Logarithmic Functions
3.3 Rules of Logarithms
3.4 Exponential and Logarithmic Equations and Inequalities
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Exercises for Calculus
Chapter 3 Practice Test A
Chapter 3 Practice Test B
4. Trigonometric Functions
4.1 Angles and Their Measure
4.2 The Unit Circle; Trigonometric Functions
4.3 Graphs of the Sine and Cosine Functions
4.4 Graphs of the Other Trigonometric Functions
4.5 Inverse Trigonometric Functions
4.6 Right-Triangle Trigonometry
4.7 Trigonometric Identities
4.8 Sum and Difference Formulas
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Exercises for Calculus
Chapter 4 Practice Test A
Chapter 4 Practice Test B
5. Applications of Trigonometric Functions
5.1 The Law of Sines and the Law of Cosines
5.2 Areas of Polygons Using Trigonometry
5.3 Polar Coordinates
5.4 Parametric Equations
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Exercises for Calculus
Chapter 5 Practice Test A
Chapter 5 Practice Test B
6. Further Topics in Algebra
6.1 Sequences and Series
6.2 Arithmetic Sequences; Partial Sums
6.3 Geometric Sequences and Series
6.4 Systems of Equations in Two Variables
6.5 Partial-Fraction Decomposition
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Exercises for Calculus
Chapter 6 Practice Test A
Chapter 6 Practice Test B
Appendix: Answers to Practice |
This whole series of books is very popular among homeschoolers. It works especially well for the student who is coming out of a school situation and needs work on foundational math. It also works well for the student who is intimidated by Saxon or other texts.
Sets include answer keys. You can also order the books without answer keys if you already have the answer keys. |
Product Description
Review
"Let me begin by saying that I really like this book, and I do not say that of very many books. What impresses me most is the level of motivation and explanation given for the basic logic, the construction of proofs, and the ways of thinking about proofs that this book provides in its first few sections. It felt that the author was talking to the reader the way I would like to talk to students. There was an air of familiarity there. All kinds of useful remarks were made, the type I would like to make in my lectures." — Aimo Hinkkanen, University of Illinois at Urbana
"The writing style is suitable for our students. It is clear, logical, and concise. The examples are very helpful and well-developed. The topics are thoroughly covered and at the appropriate level for our students. The material is technically accurate, and the pedagogical material is effectively presented." — John Konvalina, University of Nebraska at Omaha
From the Publisher
A solid presentation of the analysis of functions of a real variable -- with special attention on reading and writing proofs.
--This text refers to an alternate
Hardcover
edition.
I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book. For the self-studier, this book is excellent! I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc... For the beginner, this book is superior to all of them. A plethora of examples. Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought. If you already have had some Analysis, then this book is probably not for you. But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat. After this book, one should be able to tackle "Papa Rudin". For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis". This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration. On the whole, this is a great start! If proof-based math is new to you, then you will appreciate the first chapter on proofs. Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration. That's really only nit picking, though. |
"The best Algebra tutorial program I have seen... in a class by itself." Macworld
We start with a word problem one would never have to solve in life. Super. The answer is 2n so I choose n/2 to see what happens. The software says "Incorrect" in red, does not offer the correct answer, and simply encourages me to go on to the next slide. Very helpful. I hit Next.
Eventually we get to play with two helicopters to solve x/2 - 3 = 1. I found myself wondering where the two expressions came from and what a helicopter would be doing hovering at x/2 - 3, but I was always a troublemaker in school. Nowhere does the software talk about needing to keep the two helicoptesr at the same altitude, let alone why we have to. Me, I like see-saws which are level when the two sides are the same, just as we want to preserve an equation's truth as we transform it. Anyway...
Clicking +3 on the first guy moves it up but leaves the expression as x/2 - 3. It should have changed to x/2, the way the other guy changed from 1 to 4 when I clicked +3. Clicking x2 (meaning multiply, not the variable x) finally changes the first guy to x. The other guy continues to work and becomes 8.
Now the material simply goes wrong, saying we have to add before we multiply. No, that just makes it easier. And it gets worse: the text says that if we multiply by 2 first we will end up with the wrong answer, x=5. Nonsense, as the graphic shows: we end up with x - 3 = 5, what it calls "an incomplete solution". Thought one: well then it is not a solution! Add 3 to both sides!! And how on Earth did we get to x-3=5? By going x/2-3=1 to 2(x/2-3)=2*1 to x-6=2 to x-6+3=2+3. Right, they accepted as inevitable the two operations of adding at most 3 and multiplying by at most 3, with nothing else permitted. Hunh?
Just this little bit of material is wrong in one place, inconsistent with itself, confusing, unmotivating, and plain leaves out the fundamental concept of preserving the truth of the equation as necessary. On-line and interactive is only as good as the underlying fundamental material, and in that regard Monterey comes up short.
A blow-by-blow replay of a disappointing on-line Algebra experience at the Monterey Institute. |
Welcome to my page.
Open Algebra Textbooks
Writing these textbooks was quite the effort! It is nice to be done... Elementary and Intermediate Algebra both have been published by Flat World Knowledge
Elementary Algebra is an open textbook designed to be used in the first part of a two part algebra course. It is written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.
Intermediate Algebra offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. In addition to the creative commons license and FWK features, this textbook offers a true pedagogical improvement over the current very expensive options.
Latest Activity
I spent the summer converting my old algebra study guide to ePub format and was able to have it listed in the Google Play store.[ Click here to read my blog post about it ]It was not that hard to do and the Google Play eBook reader is quite good! It looks ok on all of my devices and transfers between them seamlessly. Downside: You have to buy this FREE ebook. Not a problem for Google Play…See MoreElementary Algebra Videos - a visually searchable playlist.Feel free to copy-and-paste these links into your LMS -- it works…See More I will update this post when I am finished.See More
I am trying to build a circle of educators interested in education technology and open educational resources. If you add me to a circle I will circle you back as soon as I can. If you are just starting out with Google+ it may seem like a ghost town at first. I suggest searching for "shared education circles" and adding a shared circle or two for an instant start. Certainly,…See More
John Redden's Blog |
Book Description: This text uses the language and notation of vectors and matrices to clarify issues in multivariable calculus. Accessible to anyone with a good background in single-variable calculus, it presents more linear algebra than usually found in a multivariable calculus book. Colley balances this with very clear and expansive exposition, many figures, and numerous, wide-ranging exercises. Instructors will appreciate Colley's writing style, mathematical precision, level of rigor, and full selection of topics treated. Vectors: Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Some n-Dimensional Geometry. New Coordinate Systems. Differentiation in Several Variables: Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newton's Method. The Chain Rule. Directional Derivatives and the Gradient. Vector-Valued Functions: Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator. Maxima and Minima in Several Variables: Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema. Multiple Integration: Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration. Line Integrals: Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields. Surface Integrals and Vector Analysis: Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations. Vector Analysis in Higher Dimensions: An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem. For all readers interested in multivariable calculus. |
math skills needed for a successful foodservice career—now in a new edition
Culinary Calculations, Second Edition provides the mathematical knowledge and skills that are essential for a successful career in today's competitive foodservice industry. This user-friendly guide starts with basic principles before introducing more specialized topics like recipe conversion and costing, AP/EP, menu pricing, and inventory costs. Written in a nontechnical, easy-to-understand style, the book features a running case study that applies math concepts to a real-world example: opening a restaurant.
This revised and updated Second Edition of Culinary Calculations covers relevant math skills for four key areas:
Basic math for the culinary arts and foodservice industry
Math for the professional kitchen
Math for the business side of the foodservice industry
Computer applications for the foodservice industry
Each chapter is rich with resources, including learning objectives, helpful callout boxes for particular concepts, example menus and price lists, and information tables. Review questions, homework problems, and the case study end each chapter. Also included is an answer key for the even-numbered problems throughout the book.
Culinary Calculations, Second Edition provides readers with a better understanding of the culinary math skills needed to expand their foodservice knowledge and sharpen their business savvy as they strive for success in their careers in the foodservice industry. |
University of Denver
Courses
Early Math and Kids Play MathMATH 0110 (3 credits)
This course is designed to introduce early childhood teachers to early mathematics and the Kids Play Math accompanying software. Teachers learn many activities that can be used to integrate early mathematics into the early childhood classroom. Implementation can occur alongside any curriculum to support increased mathematics learning and school readiness. This course is targeted for Head Start teachers in preschool settings.
Fundamentals of MathematicsMATH 0900 (2 credits)
This course covers number sets, arithmetic operations (using whole numbers, fractions and decimals), working with ratios, proportions and percents, rounding off numbers and using powers and roots of numbers. It extends to algebraic operations, solving elementary equations, the coordinate plane and graphing linear equations. It concludes with elementary logarithms and basic algebraic functions. Hand-held sceintific calculators are used throughout the course. This is a 2-non-degree-credit course. There are no prerequisites.
Elements of College AlgebraMATH 1010 (2 or 4 credits)
This course is designed to review the required algebra skills to be successful in MATH 1200. The students study the following topics: review of basic algebra, solving equations and inequalities, rectangular coordinate systems and graphing, polynomial and rational functions, exponential and logarithmic functions, and solving exponential and logarithmic equations.
Mathematics for BusinessMATH 1050 (4 credits)
Foundations SeminarMATH 1150 (4 credits)
The seminars offer challenging and interesting mathematical topics that require only high school mathematics. Examples of seminars are Introduction to Crytography, Patterns and Symmetry, Mathematical Art and Patterns of Voting.
Statistical ReasoningMATH 1160 (4 credits)
This course serves as an introduction to the fundamental concepts in statistics and probability as they apply to the social sciences. The course emphasizes statistical reasoning as it applies to decision-making, the use of probability in thinking about and solving problems, and the interpretation of results. Topics include sampling theory, presenting data using tables, charts and graphs, summarizing and describing data with numerical measures, fundamentals of probability, discrete and continuous probability distributions, the normal probability distribution, sampling distribution, estimation, confidence intervals, hypothesis testing, and regression and correlation. Prerequisite: MATC 1100.
Calculus - Business & Soc SciMATH 1200 (0 or 4 credits)
This is a one-quarter course for students in business, social sciences, and liberal arts. It covers elementary differential calculus with emphasis on applications to business and the social sciences. Topics include functions, graphs, limits, continuity, differentiation, and mathematical models. Students are required to attend weekly labs.
College Algebra & TrigonometryMATH 1750 (4 credits)
Selected topics in algebra and analytic trigonometry intended to prepare students for the calculus sequence (MATH 1951, 1952, 1953). Cannot be used to satisfy the Analytical Inquiry: The Natural & Physical World requirement.
Calculus IMATH 1951 (4 credits)
Limits, continuity, differentiation of functions of one variable, applications of the derivative. Students with high school trigonometry should enter th Calculus sequence in fall quarter. Others should complete prerequisite MATH 1750 and enter the Calculus sequence in winter quarter. Prerequisite: MATH 1750 or equivalent.
Calculus IIMATH 1952 (4 credits)
Differentiation and integration of functions of one variable especially focusing on the theory, techniques and applications of integration. Prerequisite: MATH 1951.
Calculus IIIMATH 1953 (4 credits)
Honors Calculus IIMATH 1962 (4 credits)
Same topics as MATH 1952 treated rigorously and conceptually. Topics include differentiation and integration of functions of one variable especially focusing on the theory, techniques and applications of integration. Prerequisites: MATH 1951 and permission of instructor.
Honors Calculus IIIMATH 1963 (4 credits)
Same topics as MATH 1953 treated rigorously and conceptually. Topics include integration of functions of one variable, infinite sequences and series, polar coordinates, parametric equations. Prerequisites: MATH 1952 or MATH 1962 and permission of instructor.
Study Abroad Resident CreditMATH 1988 (0 to 18 credits)
Directed StudyMATH 1992 (1 to 10 credits)
Symbolic LogicMATH 2050 (4 credits)
Modern propositional logic; symbolization and calculus of predicates, especially predicates of relation. Cross-listed with PHIL 2160.
Elements of Linear AlgebraMATH 2060 (4 credits)
Matrices, systems of linear equations, vectors, eigenvalues and eigenvectors; idea of a vector space; applications in the physical, social, engineering and life sciences. Prerequisite: MATH 1750 or equivalent.
Study Abroad Resident CreditMATH 2988 (0 to 18 credits)
Directed StudyMATH 2992 (1 to 10 credits)
The Real World SeminarMATH 3000 (1 credits)
Lectures by alumni and others on surviving culture shock when leaving the University and entering the job world. Open to all students regardless of major. Cross-listed with COMP 3000.
History of MathematicsMATH 3010 (4 credits)
This course surveys major mathematical developments beginning with ancient Egyptians and Greeks and tracing the development through Hindu-Indian mathematics, Arabic mathematics, and European mathematics up to the 18th century. Prerequisite: MATH 1953.
Graduate SeminarMATH 4300 (1 to 4 credits)
Students research a topic of their choosing with the aid of a faculty member, and then prepare and present a formal lecture on the subject. Prerequisite: graduate standing or consent of the instructor. |
Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate cour... read more
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Elements of Abstract Algebra by Allan Clark Lucid coverage of the major theories of abstract algebra, with helpful illustrations and exercises included throughout. Unabridged, corrected republication of the work originally published 1971. Bibliography. Index. Includes 24 tables and figures.
Abstract Algebra by W. E. Deskins Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.
Introduction to Proof in Abstract Mathematics by Andrew Wohlgemuth This undergraduate text teaches students what constitutes an acceptable proof, and it develops their ability to do proofs of routine problems as well as those requiring creative insights. 1990 edition.
Theory of Sets by E. Kamke Introductory treatment emphasizes fundamentals, covering rudiments; arbitrary sets and their cardinal numbers; ordered sets and their ordered types; and well-ordered sets and their ordinal numbers. "Exceptionally well written." — School Science and Mathematics.
Basic Algebra II: Second Edition by Nathan Jacobson This classic text and standard reference comprises all subjects of a first-year graduate-level course, including in-depth coverage of groups and polynomials and extensive use of categories and functors. 1989 edition.
Mathematical Physics: A Popular Introduction by Francis Bitter Reader-friendly guide offers illustrative examples of the rules of physical science and how they were formulated. Direct, nontechnical terms explain methods of fact gathering, analysis, and experimentation. 60 figures. 1963 edition.
Linear Algebra by Georgi E. Shilov Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, and more.
Product Description:
Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math majors and future math teachers. This second edition features additional exercises to improve student familiarity with applications. An introductory chapter traces concepts of abstract algebra from their historical roots. Succeeding chapters avoid the conventional format of definition-theorem-proof-corollary-example; instead, they take the form of a discussion with students, focusing on explanations and offering motivation. Each chapter rests upon a central theme, usually a specific application or use. The author provides elementary background as needed and discusses standard topics in their usual order. He introduces many advanced and peripheral subjects in the plentiful exercises, which are accompanied by ample instruction and commentary and offer a wide range of experiences to students at different levels of ability |
Course in Mathematical Statistics, Second Edition, contains enough material for a year-long course in probability and statistics for advanced undergraduate or first-year graduate students, or it can be used independently for a one-semester (or even one-quarter) course in probability alone. It bridges the gap between high and intermediate level texts so students without a sophisticated mathematical background can assimilate a fairly broad spectrum of the theorems and results from mathematical statistics. The coverage is extensive, and consists of probability and distribution theory, and statistical inference. |
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.
A curve fitting program: Lorentzian, Sine, Exponential & Power series are available models to match your data. A Lorentzian series is recommended for real data especially for multiple peaked data. Another improved productivity example.
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.
Home Construction Estimator is a full-featured program for providing fast, easy and accurate building estimates on residential home construction. Makes home construction estimating as easy as answering a few questions.
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. |
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College Algebra and Trigonometry: A Graphing Approach
Intended to provide a flexible approach to the college algebra and trigonometry curriculum that emphasizes real-world applications, this text ...Show synopsisIntended to provide a flexible approach to the college algebra and trigonometry curriculum that emphasizes real-world applications, this text integrates technology into the presentation without making it an end in itself, and is suitable for a variety of audiences. Mathematical concepts are presented in an informal manner that stresses meaningful motivation, careful explanations, and numerous examples, with an ongoing focus on real-world problem solving. Pedagogical elements including chapter opening applications, graphing explorations, technology tips, calculator investigations, and discovery projects are some of the tools students will use to master the material and begin applying the mathematics to solve real-world problems. CONTEMPORARY COLLEGE ALGEBRA AND TRIGONOMETRY includes a full review of basic algebra in Chapter 0 and full coverage of trigonometry to prepare students for the standard science/engineering calculus sequence. (The companion volume, CONTEMPORARY COLLEGE ALGEBRA includes all of the non-trigonometry topics, covered in sufficient detail to prepare a student for a business/social science calculus course.) Those who are familiar with the author's CONTEMPORARY PRECALCULUS should note that this book covers topics in a different order, and with a slower, gentle approach. Also, more drill exercises are included |
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