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Cecilton Microsoft ExcelAlgebra 1 can be very confusing when it is not explained properly. It is not a hard subject matter, it just demands a good teacher. I use basic formulas and equations in algebra 1 on a daily basis, and believe this can help me to help you understand the subject matter using real world examples.
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Math
The goal of Alabama's K-12 mathematics program is to empower all students to live and work in the twenty-first century with the mathematical skills, understandings, and attitudes they will need to be successful in their careers and daily lives. Mathematically empowered students are flexible and resourceful problem solvers who understand and value mathematics and communicate ideas effectively. Educators, using the Alabama Course of Study: Mathematics as a basis for curriculum development and instructional decision-making, provide opportunities that enable all students to use mathematics in everyday life and in the workplace.
This course of study specifies a minimum foundation of mathematics to be learned by all students, including students with disabilities. Content standards are included for each grade level and course. These standards are aligned to build upon each other across the grades without repetition. School systems are encouraged to expand the content standards when appropriate to address the needs of their students.
The recommendations of the Principles and Standards for School Mathematics (PSSM) from the National Council of Teachers of Mathematics (NCTM) are incorporated into the conceptual framework, position statements, and content standards of this course of study. The content in each grade level and course is organized using the five PSSM content standards. These five content standards that serve as strands in this document are Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability. The PSSM process standards of Problem Solving, Reasoning and Proof, Communication, Connections, and Representation should be integrated into instruction as outlined in PSSM.
In order to effectively implement this document, local educators must use this course of study to develop local curriculum guides or local courses of study. Implementation of the Alabama Course of Study: Mathematics is an important step in providing students with a solid foundation of knowledge, skills, and understanding in mathematics. This foundation is an essential element in leading students toward mathematical empowerment, thereby enhancing their opportunities and options for the future.
Think Central
Student eBooks are fun and interactive digital versions of the student's traditional print textbook used in the classroom. Every day, at home or at school, students will be able to log into ThinkCentral and immediately be able to view and explore the reading & math materials.
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This is a free, online book that was originally published as a university lecture series in 1992. Chapters include the following: 1. Mathematical model 2. First integrals of boundary motion 3. Algebraic solutions 4. Contraction of a gas bubble 5. Evolution of a multiply connected domain 6. Evolution with topological transformations 7. Contraction problem on surfaces.
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Calculus: Understanding Its Concept and Methods is a complete electronic textbook featuring live calculations and animated, interactive graphics. It uses the included Scientific Notebook® program to display text, mathematics, and graphics on your screen and to provide an interactive environment including examples with user-defined functions, animations, and algorithmically generated self-tests. This environment encourages a focus on mathematical problem solving, experimentation, verification, and communication of results.
This electronic book covers the content normally taught in a three-semester calculus sequence. The material is presented in a way that encourages mathematical problem solving, experimentation, exploration, and communication. It includes explanations, examples, explorations, problem sets, self tests, and resource information. Many of the files contain animations that you can manipulate and control. Other files are interactive: you can define your own function, or change some parameters, and observe the results. This allows you to achieve important insights from specific examples.
Calculus: Understanding Its Concept and Methods is thoroughly indexed and hyperlinked to provide easy access to relevant information. It is appropriate for independent study, as a supplement to any standard calculus text, or for distance learning.
Calculus is the mathematics of change and approximation. With the computer algebra system in Scientific Notebook®, you can interactively explore examples and carry out experiments. The skills you develop will help you to solve problems you encounter in the future because you are always dealing with natural mathematical notation and general problem-solving methods.
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Algebraic Geometry This is a first course in commutative algebraic geometry, for
students
who have some background in basic algebra, and in particular in the
theory of rings and fields. The first part covers some basic
commutative algebra, building towards
the basic theory of affine and projective algebraic varieties.
Learning
Outcomes:On successful
completion of this module, the student
will: 1. be familiar with the basic
concepts, methods and results of elementary Algebraic Geometry and
elementary Commutative Algebra. 2. be able to recognise affine
and projective algebraic varieties, describe them them in the language
of Commutative Algebra and analyse their basic
properties using the methods and tools of the later. 3. be able to describe, construct
and analyze affine and projective algebraic varieties, including their
singular points. 3. be able to apply the
techniques of ideal theory to basic problems in the theory of affine
and projective algebraic curves and surfaces. 4. be able to apply the
techniques of basic elimination theory to problems in elementary
algebraic geometry 5. be familiar with the basic
concepts of classical intersection theory.
Core Textbooks:
1. Rings and ideals; definition of prime and
maximal
ideals;
multiplicative sets; quotient of a ring by an ideal; integral domains
and principal ideal domains (P.I.D.s).
2. Ideal generated by a set; ideal intersections and ideal sums.
3. Characterizations of prime and maximal ideals. The prime and maximal
spectra of a ring. Krull's theorem; the Krull dimension of a ring.
Noetherian rings. Ideal-theoretic characterization of fields and
integral domains.
4. Basic theory of semilattices and order lattices. The lattice
($J(R),+,\cap)$ of ideals of a ring. The
product of ideals. The semiring $(J(R),+,\cdot)$. 5. Radical of an ideal; roots of an
element;
nilpotent elements; the
nilradical of a ring. Reduced rings. Radical ideals. The quotient
criterion for radical ideals. Prime ideals are radical. The reduction
of a ring. I-order of elements and order of nilpotency. The
radicalization map as a closure operator. The semilattice of radical
ideals.
6. Closure and kernel operators.
7. Modules and algebras over a commutative ring; morphism and
isomorphism of algebras; monoid algebras and polynomial algebras;
basic terminology and conventions for polynomials. Z-modules are
the same as Abelian groups; Z-algebras are the same as commutative
rings. First isomorphism theorem for commutative R-algebras;
application to commutative rings.
8. Finitely generated algebras. Presentation of a finitely-generated
algebra through generators and relations; polynomial algebras are the
free commutative algebras on finite sets of generators; the
universality property of polynomial algebras.
9. Galois correspondences; examples.
10. Hilbert's basis theorem.
The case of coefficients in a field. 10. Hilbert's weal and hard nullstellensatz. The (affine) ideal
variety-correspondence.
11. Affine varieties; the Zarsiski tangent space; singular points.
12. Projective space; homogeneous polynomials; projective varieties and
projectivisation.
Tutorials:
Tutorial 1: The ideals of the ring of integers
--part 1:
-The ring of integers is a P.I.D.
-Divisibility as a partial order relation;
connection with the inclusion relation on ideals
-Ideal intersections and sums correspond to the lcm
and gcd.
Tutorial 2: The ideals of the ring of integers -- part 2:
-The lattice J(Z) and the spectrum Spec(Z). Meaning
of Noetherianness of Z.
-Irreducibility versus primeness. Primality.
Primality implies that all irreducibles are prime.
-The Euclidean algorithm and the unique
factorization theorem (the fundamental theorem of arithmetic).
-Basics on generalizations: Arithmetic in Euclidean
Rings, P.I.Ds, Bezout domains, UFD's, GCDs, (pre-)Schreier domains etc.
Tutorial 3: Square-free content of a natural number; radicalization map
and radical ideals of Z; the nilradical of the rings Z_n;
description of the nilpotent elements of Z_n.
Tutorial 4: Viete's relations.
Tutorial 5: Resultants and discriminants. Revision Tutorial: Affine curves; computing intersection points
through resultants; some examples of intersection multiplicity;
ordinary double points and cusps for affine real and complex curves.
Some lecture notes(with
apologies for any typos):
The exam will consist of 4 questions (each worth 25 points), of which
you have to answer 3 questions correctly in order to get a maximum
score (a score of 75 points will be renormalized to 100%). For those
questions which have sub-questions, each sub-question will be numbered
and contain an indication of how many points it is worth. For example:
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In continuation with the topics studied in Algebra I, it will develop the real number system and will include a study of the complex numbers as a mathematical system. Students will study the ideas of relations and functions and expand the concept of functions to include quadratic, square root, exponential and logarithmic functions, and rational numbers. Emphasis will also be placed on the analysis of conic concepts with labs and the development of additional real life problem solving skills and applications. Emphasis will be placed on the application of concepts and skills introduced in Algebra II. The level of instruction/curriculum will focus on preparing the student for further advanced placement courses.
I will be available for tutoring Tuesdays and Thursdays after school from 3:00 to 4:00.Any changes to tutoring will be announced in class and on the class website.If the posted tutorial times do not work in your schedule be sure to discuss alternatives with me as soon as possible.Other math teachers are also available at other times which are posted in the math hallway.
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This quantitative reasoning text is written expressly for those students, providing them with the mathematical reasoning and quantitative literacy skills they'll need to make good decisions throughout their lives.
Common-sense applications of mathematics engage students while underscoring the practical, essential uses of math..
For more information about the title Using and Understanding Mathematics: A Quantitative Reasoning Approach (4th
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25. Convergence to Axiomatic View
Ends and Values - a matter of choice
The ends and values of mathematics education, or logical and quantitative
skill development may vary between students. Teachers who have been
suddenly assigned a mathematics courses, despite a lack of background in
mathematics or a quantitive discipline, may not value the logical
development of mathematics. For many students and teachers, mathematics
appears to be collection of facts and methods to learn and teach without
any attempt to obtain or provide a thought-based development.
One grade 8 student on observing my attempts to explain and justify
mathematical methods instead of teaching them as facts told me that
mathematics teachers were hired only to present mathematical correct
methods, and thus I was doing my job properly, the methods I was covering
would not need explanation nor justification. For many students, the
notion that mathematics has a logical structure, one in which ideas are
developed and derived in place of being given, is odd and not necessary.
Indeed, they may be partially correct. The common know-how in mathematics
may be met and mastered with comprehension or thought-based development,
but for students or the common person in the streets, the take-home value
of an operational command of counting, figuring and measuring skills with
take home value in a repeatable and reproducible manner is more important
than full or partial comprehension of why methods work in the first years
of mathematics education and for students who may or may not go further
in mathematics.
The site approach and Choice
Logic chapters 1 to 5 in Three Skills for Algebra end with a discussion
of Islands and Divisions of Knowledge. The latter provides a metaphor
for the organization of mathematics and the possibility of having
different starting points for its development.
Site pages show with two or more paths how the existence of real numbers
and their arithmetic properties can be derived from common practices
assumptions about numbers and geometry with maps and plans. The
demonstrations appear to be empirically and pedagogical sound, given the
need to introduce skills, patterns and even axioms in an inductive
manner. Site pages also provide a systematics introduction to algebra, or
the shorthand role of letters and symbols.
Nostalgia or attraction to the rigour of modern mathematics means the
demonstration were written in a thought-based manner with as much rigour
as possible. But there is a difficulty. Too much explanation may
overwhelm skill mastery. Moreover, mastery of skills with care to avoid
the domino effect of errors has great take-home value in which full or
partial comprehension of why is optional. The foregoing suggests ends and
values for instruction that support a rigourous development of skills,
step by step, because of the take-home value with explanations why being
available and present where they do not overwhelm.
Site pages are part of a two level approach POMME. The first level and
part of the second are dedicated to providing skills and concepts with
take-home value, by rote if need-be. The second level, what is left, is
dedicated to a thought-based development that does not begin with the
modern mathematics mid-way axioms for secondary mathematics, but implies
them. See site slow paths, computational and geometric, for the
thought-based development of numbers and their properties from counting
to the properties of real and complex numbers. The paths may not be given
in classes where students have mixed ends and values - some wanting
mathematics with take-home value only - some wanting to continue onto
college programs in disciplines requiring or best taught with a command
of calculus. The second level in full, as presented here, values thought-
or pattern-based development of skills and concepts as possible
preparation for college studies in mathematical fields.
The thought-based development of numbers and their properties from
counting to the properties of real and complex number, with the
subsequent assumption of those properties as axioms for the further
logical development of mathematics implies a partial convergence of the
site two level approach POMME for quantitative and logical skill
development with modern mathematics curricula.
The modern mathematics curricula I saw began well at the start of senior
high school mathematics, but soon departed from pure mathematics with the
employment of a diagrams in the introduction of trigonometry, analytic
geometry, and calculus to develop methods and prove theorems. The
diagram-free development, a possibility in university mathematics, would
be too difficult and have no context in senior high school mathematics.
Whence some departure is needed - in for a penny, in for pound. The site
development provides a departure in a two-level manner, with one level
focusing on empirical rigour in skill mastery and the second level offer
a thought-based development consistent first the need to sanction and
extend common skills and know-how, with numbers, maps and plans.
Modern Mathematics Curricula,
In the modern mathematics curriculum, circa 1955-1990, the existence of
the real numbers and the satisfaction of above properties were given as
assumptions or axioms. That provides a simple starting point for a
logical development of secondary and college mathematics. A justification
of the axioms might then be seen by students who enter mathematics
studies in university. In particular, assumptions for set existence and
"safe" set construction provide an axiomatic codification, Euclidean
style, for pure mathematics. For rigour, the approach sould be context-
and diagram-free, a rigour not possible before university level studies
in pure mathematics. As said, the modern mathematics curricula depart
with the employment of diagrams in the introduction of trigonometry,
analytic geometry, and calculus to develop methods and prove theorems.
For all students, and many teachers, axioms for real numbers and within
them, rational numbers, integers, natural numbers and whole numbers, the
axioms will appear and will have to be accepted without explanation. But
the axioms were not chosen to continue and sanction common knowledge and
practices with decimals and diagrams which would have had take-home
value. The axioms for real numbers provided a view of numbers that did
not explicitly sanction and support common skills in counting, figuring
and measuring with maps, plans and decimals. The modern mathematics
curricula was not designed to meet the needs of students who would have
benefited from mathematics with take-home value. The modern mathematics
curricula was designed to prepare students for college programs that
required calculus or beyond, with context-free development being an
objective. Axioms and further development of mathematics did not sanction
earlier number skills and sense with fractions and decimals.
For many students and many of their teachers, the modern mathematics
curricula was further flawed in that the secondary level axioms were
described algebraically with out a systematics introduction of the
shorthand role of letters and symbols. Whence the deductive axiomatic
development of mathematics was beyond the reach of students and teachers
for whom the algebraic way of reasoning with letters and symbols on paper
was not a natural talent. The slower and more detailed systematic
development of algebraic reasoning in site pages points to a remedy, one
that requires less natural talent
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for Economists1 Introduction1.1 Motivation Why do we need to know mathematics in order to learn economics? What is economics? In economics we learn how the economy works in various situations. An economy consists of various people (consu
6 Integration (A.4)6.1 Indenite Integral Consider a continuous function f (x), where f (x) > 0 for all x. Consider the area under the graph of y = f (x) from a certain point a to anotherpoint x and denote it by A(x; a). What is the derivative of A(x;
Denition 36 (p.161). An m m matrix A = (ai j ) is called an upper-triangular matrixif ai j = 0 for i > j. A is called a lower-triangular matrix if ai j = 0 for i < j. A is calleda diagonal matrix if ai j = 0 for i = j.Theorem 56 (Fact 26.11, p.731). Th
Stat 351 Fall 2007Assignment #9This assignment is due at the beginning of class on Friday, November 30, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of exposit
Statistics 351 (Fall 2007)Review of Linear AlgebraSuppose that A is the symmetric matrix1 1 0A = 1 2 1 .013Determine the eigenvalues and eigenvectors of A.Recall that a real number is an eigenvalue of A if Av = v for some vector v = 0. We callv a
Statistics 351 Midterm #1 October 10, 2007This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate yo2 November 16, 2007This exam is worth 50 points.There are 5 problems on 5 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in orde
Stat 351 Fall 2007Assignment #1This assignment is due at the beginning of class on Monday, September 10, 2007. You must submitall problems that are marked with an asterix (*).1.* Send me an email to say Hello. If I have never taught you before, tell
Stat 351 Fall 2007Assignment #5This assignment is due at the beginning of class on Friday, October 5, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of expositio
Stat 351 Fall 2007Assignment #7This assignment is due at the beginning of class on Friday, November 9, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of expositi
Statistics 351 Midterm #1 October 18, 2006This exam has 4 problems and 5 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate yo
Statistics 351 Midterm #2 November 17, 2006This exam is worth 40 points.There are 5 problems on 5 numbered pages. You may attempt all ve and yourfour highest scores will be taken as your mark. You might want to read all vequestions before you begin.Y
Statistics 351 Fall 2006 (Kozdron) Midterm #2 Solutions1. (a) Recall that a square matrix is strictly positive denite if and only if the determinantsof all of its upper block diagonal matrices are strictly positive. Since2 22 3=we see that det( 1 )
Art-labeling Activity: Figure 21.15Part ADrag the appropriate labels to their respective targets.This content requires Adobe Flash Player 10.0.0.0 or newer.ANSWER:ViewCorrectIP: Class I and Class II MHC ProteinsClick on the link or the image below
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Nelson Functions 11 provides 100% coverage of the NEW Ontario curriculum for Grade 11 University (MCR 3U) while preparing students for success in Grade 12 and beyond. Key Features & Benefits include:
• Skills and Concepts Review at the beginning of every chapter
• Multiple solved exam...
Nelson Principles of Mathematics 10 ensures students build a solid foundation of learning so they are prepared for success in senior level courses. The program supports the diverse needs of students (through multiple entry points to help a varying range of learners), and offers extensive supp...
Big Ideas from Dr. Small provides math teachers with what they need to know to teach the curriculum while focusing on the big ideas for each math concept. Each book includes hundreds of practical activities and follow-up questions to use in the classroom. The accompanying Facilitator's Guide ...
The Mathematics Teacher eMentor DVD is a flexible, interactive professional learning resource that brings the expertise of leading Canadian math educator, Dr. Marian Small, to teachers across Canada.
With DVDs for K-3, Grades 4-6, and Grades 7-9, the Mathematics Teacher eMentor provides the full s...
More Good Questions, written specifically for secondary mathematics teachers, presents two powerful and universal strategies that teachers can use to differentiate instruction across all math content: Open Questions and Parallel Tasks. Showing teachers how to get started and become expert ...
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Mrs. Paula Smith Algebra 2
This course is mainly a Junior level course. We take the topics learned in Algebra 1 and raise them to the next level. To me, Algebra 2 is all about graphing. First semester, we cover lines and their equations, as well as matrices and exponents. Second semester is more rigorous, as we dive into radicals, exponentials, logarithms, sequences and series.
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and 12 subjects for the Victorian Certificate of Education (VCE) under guidelines in a. "Study Design" ...methods of calculation is critical for all branches of Mathematics. ... To increase congruence between real maths and school maths. 3. To achieve deeper learning by students. 4. .....Essentials of Mathematics Education
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HOMEWORK
At least 2/3 of the questions on the midterm and final exams
will be similar to homework problems, so doing them is
essential in order to do well in the course. There are
two types of homework problems, WeBWorK Assignments and
Suggested Problems from the text as detailed below. WeBWorK
assignments are worth 10% of your grade. Your instructor may
give short biweekly quizzes in class based on the WeBWork assignments
and suggested problems that are worth another 5%. See your section website for details.
WeBWorK Assignments: Weekly homework assignments using
the WeBWorK online homework system will be posted at least one
week before their due date. These assignments will be worth 10%
of your course grade. You will need to use a Web browser to access
these assignments and submit your answers. You must submit your
answers before the due date, which will be 9 p.m. on Fridays.
Necessary material will be covered in class by early in the
week. It is strongly recommended that you work on these
assignments well in advance of the due date. Questions are randomized,
so different students will receive different versions of the problems.
To access your particular WeBWorK assignments, click here and login using your CWL or click the MATH101_ALL_2012W2
link at If you prefer, you can also access these assignments by logging in to Connect at Note that WeBWorK assignments are the only element of MATH
101 available on Connect.
Suggested Problems: Additional problems from the text
will be posted below. These problems are mostly odd-numbered problems
from the text. Answers to the odd-numbered problems are given
in the back of the text, and solutions to these problems are in
the Student Solutions Manual, which is included with the
text in the package sold by the UBC Bookstore. These problems
are not collected.
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PCI Ed's Algebra City Covers 28 Common Gaps in Student Understanding
By Dian Schaffhauser
06/26/12
PCI Education, a company that develops curriculum products, has published a set of learning materials specifically for kids in grades eight through 10 having trouble with Algebra 1 concepts. "Algebra City" is a set of four workbooks that address 28 common algebraic misconceptions using a graphic novel approach and web-based practice problems. According to the company, the program is intended to be used not for standard curriculum but for intervention, pinpointing areas where students are struggling.
The materials consist of four student editions, each one covering seven of the 28 topics, as well as a teacher set with an assessment CD, a teacher resource CD, and access to the interactive activities.
An ExamView Assessment Suite includes pre- and post-tests for the program, the book, and individual unit levels, as well as an item bank and test generator, and reporting features.
"Too often, students struggle to learn critical algebra skills they need both inside and outside the classroom," said Lee Wilson, president and CEO of PCI Education. "Algebra City is targeted intervention that encourages students to reconnect to algebra in one or more areas of misunderstanding, while allowing teachers to leverage the investment in their core algebra curriculum."
A classroom starter pack is priced at $599.95 and includes the teacher's kit and a five-pack of student editions, which has five copies of each of the four books in the series
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Differential equations
This unit extends the ideas introduced in the unit on first-order differential...
This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equations which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and some familiarity with complex numbers.
After studying this unit you should:
be able to solve homogeneous second-order equations;
know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;
know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.
Contents
Differential equations
Introduction
This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equation which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and have some familiarity with complex numbers
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This resource looks at three detailed case studies in the form of student projects which clearly illustrate how mathematics and geography can be effectively linked. This booklet serves as good preparation material for CPD work between departments.
Food and settlement
This case study looks at Nomads and talks about the areas booklet is primarily aimed at the mathematics teacher, but should also be of interest to teachers of science. It sets out a number of case studies suitable for mathematical modelling.
The book starts with an explanation of the mathematical modelling process then suggests specific areas of study which include:
Forecasting: booklet is primarily aimed at the mathematics teacher, but should also be of interest to teachers of science. It sets out a number of case studies suitable for mathematical modelling with calculus. The book starts with an explanation of the mathematical modelling process then suggests specific areas of study which include:
King…
This resource looks at three areas of the physics curriculum where functions are used, creating an opportunity for cooperation between physics and mathematics teachers. This booklet serves as good preparation material for CPD work between departments.
Hooke's Law and the idea of a function
Part one explores the data produced…
This booklet is unusual in this series as it develops one theme, genetic inheritance , to show the use of mathematical ideas and terminology in science and serves as good preparation material for CPD work between departments.
The areas covered are: the inheritance of a single character, the inheritance of two characters, the genetics…
This resource looks at areas of the chemistry curriculum where mathematics is used, creating an opportunity for co-operation between chemistry and mathematics teachers. This booklet serves as good preparation material for CPD work between departments.
Section One considers how proportion is used in chemical equations, the need…
This Nuffield Foundation publication was prepared to help students master the calculations involved in GCSE Science courses. The book is divided up into a series individual topics. Each topic is presented in three parts.
• A summary of the ideas students need to know, including any important formulas.
• Worked examples,…
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0132424339
9780132424332
Elementary Statistics: For algebra-based Introductory Statistics courses. Offering the most accessible approach to statistics, with a strong visual/graphical emphasis, this book offers a vast number of examples on the premise that students learn best by "doing". The fourth edition features many updates and revisions that place increased emphasis on interpretation of results and critical thinking in addition to calculations. This emphasis on "statistical literacy" is reflective of the GAISE recommendations. «Show less
Elementary Statistics: For algebra-based Introductory Statistics courses. Offering the most accessible approach to statistics, with a strong visual/graphical emphasis, this book offers a vast number of examples on the premise that students learn best... Show more»
Rent Elementary Statistics 4th Edition today, or search our site for other Farber
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Pages
Tuesday, April 3, 2012
How Do You Teach Proof?
In the late 90's, at a different college, I taught Linear Algebra a few times. I wasn't satisfied with my teaching. I could see that the students were struggling – they really couldn't do proofs – and I had no idea how to help them.
When I decided to teach Linear this semester, I spent a lot of time beforehand studying the material. After a 12-year gap, I knew I needed to refresh my understanding. Although I still wasn't sure exactly how I would help my students learn to prove theorems, I trusted that I'd be able to figure it out.
Just like my former students, my current students have struggled with proof. One thing that has helped is the true-false questions Lay provides in every section. I often have my students vote: "Just guess, it's ok to guess wrong. Then we'll discuss it." We don't have clickers, but they're pretty willing to do it, and I keep finding out how much harder the material is for them than I had expected.
I use the true-false questions to make quizzes too. All my students aced the first quiz (on consistent systems and general solutions, no proofs), so I knew I had a great group. I made the second quiz a bit harder (1 question: does the span of two given vectors include a third vector given?). They still did pretty well, so on their third quiz I asked them to 'prove or disprove' one statement. Most of the class failed that quiz, so I gave them an alternate version the next day. They still mostly failed. I made a third version a few days later, and they finally improved.
On their first test, the 'prove or disprove' question had the lowest success rate of all the questions, but many of them did get it. Most of my students are used to acing their math classes, so I find myself reassuring them that this is a journey, and that I trust that they'll get good at this before the course is over.
I think there's one other big difference, though it's hard to pinpoint it. The work I've done over the past 4 years, working on math that challenges me, and writing about mathematics, has made me more of a mathematician, and I'm sure that's helping me teach this course better. I have Bob and Ellen Kaplan, Amanda Serenevy, and a few other great teachers at the Summer Math Circle Institute to thank, along with Josh Zucker and Paul Zeitz who work with the Bay Area math circles. I also have my blog readers to thank for motivating me to keep writing here. Thank you all!
This class is the most exciting class I've taught in a long time. For the first time in my life, we're ahead of the schedule I've set myself. That's how good my students are. And I'm getting to talk with students about what it means to do mathematics. I think they're learning something new, and I'm grateful to be a part of that.
9 comments:
It might be good to go through a quick refresher/introduction to formal logic. I'm not convinced that this is an area of the curriculum that is well-taught, and I can see how students can breeze through non-proof based classes, having experienced only if-and-only-if statements (e.g. Pythagorean Theorem) and not if->then statements (e.g. Fermat's Little Theorem).
This page has some nice simple examples as well as how to reframe the questions to tap into a more easily accessed social contract framework:
Here's a comment from Ben, who was having trouble with Blogger's anti-robot verification words. I hope people aren't giving up on commenting because of those...
In the present context I disagree with Hao's formal logic advice. I'm for this in the big picture of K-16 math education (I would aim for late elementary / early middle school, and my personal choice of text would be Raymond Smullyan's amazing puzzle books), but in a 1-semester college class on a different topic I think it won't support the goals of the class unless it takes up so much time it displaces those goals.
Here's my 2 cents. Not a silver bullet but I believe a necessary condition: first of all, make a proof exercise part of the HW regularly if you're not already doing that. Second and very important, in all cases of proofs (on HW, tests and in class when you present a proof or have them develop it), do everything you can to generate legitimate uncertainty about what the outcome is going to be. The worst thing that ever happened to the teaching of proof IMHO is that it got disconnected from the process of students actually finding out what the truth is. (For this reason I dig your "prove or disprove" typed questions, but I'm saying also approach all proofs in class from this same "not actually sure what the answer is" point of view.)
Example: recently I was working with a tutoring student on linear algebra. The student has very little proof background. Before defining a vector space I showed him a long list of examples of vector spaces including the usual R^2 and R^3 but also the set of solutions to a certain differential equation, the set of sequences with limit 0, the set of continuous functions. Later, after developing the definition, I gave him an exercise, "In a vector space is it always true that 0 times a vector is the zero vector? Or could it ever be something else?" He came back with a proof (basically just a direct calculation) that in the case where the vector space looks like column vectors, 0v=0 always; but he didn't initially recognize the limitation of his proof. I referred him to our list of vector spaces to say, "okay, you have convinced me that this will always happen for these first two items on the list, but not for the rest of the items. What about them?" He saw that at this point he legitimately didn't know whether 0v=0 would always be true in the other vector spaces. This feeling of legitimate uncertainty made the next phase of our instruction go very very smoothly. I presented a proof that referred only to the defining properties of a vector space, and now the proof was actually the reason for him to believe the result. Then when I asked him a related question (can there be a second vector v that shares with the zero vector the property that w+v=w for all w?), he was able to come up with a good proof with very little support; and he came up with a proof for the third question I asked by himself.
Again, not a magic bullet. The point is just that I think it's incredibly important for learning how to create proofs, that you actually use proofs to know what's true. The closer connected the process of searching for proofs is to the process of finding out what the truth is, the better.
Hi Professor Sue, Its me Calvin from Muskegon again ( the hotel clerk from 2011 and former student of yours). Just glad to see you alive and vigorous. Sorry that I have not been in touch, but sorrow grapes bring upon good wine so to speak. When it comes to the late 90's I do remember you embracing a technique that helped myself and other students quite well. I Look forward to seeing you again this year. If you cannot journey back to Michigan for the conference in August then I hope you all the best. You have always been on my mind since we last met instructor/Professor
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This course is an in-depth study of functions and a review of algebraic, geometric, and trigonometric principles and techniques. Students investigate and explore the characteristics of linear, polynomial, and trigonometric functions, and use graphing calculators to solve and evaluate various functions, equations, and inequalities.
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a bunch of probabilty/stats books, some macro books, a few math texts (I noticed analysis and measure theory). Tons of stuff that we're all going to be expected to be familiar with come summer math camp and first-year sequence in fall.
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Mathematics for Elementary School Teachers , 1st Edition
ISBN10: 0-538-49363-1
ISBN13: 978-0-538-49363-5
AUTHORS: Fierro
Mathematics for Elementary School Teachers is designed to give you a profound understanding of the mathematical content that you are expected to know and be able to teach. The chapters integrate the National Council of Teachers of Mathematics (NCTM) Standards and Expectations and the new Common Core State Standards, as well as research literature. The five NCTM Process Standards of problem solving, reasoning and proof, communication, connections, and representation highlight ways that teachers present content, the ways that students learn content, and various ways that students can demonstrate procedural and conceptual understanding. The worked examples and homework questions provide prospective elementary school teachers with opportunities to develop mathematical knowledge, understanding, and skills that they can apply in their own classrooms effectively. The learning path begins with the "Where Are We Going?" Chapter Openers, worked Examples with Yellow Markers that indicate the Process Standards throughout the text, to the Concept Maps, to the Section Question Sets with their "refreshers" of Process Standards, to the Chapter Organizers with Learning Outcomes and a list of the corresponding Review Questions, and finally, conclude at the Chapter Tests with their overarching Learning Outcomes
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Math Workshop
The PPC (Pre-pre-calculus) program is the component of the Math Workshop which reviews your high school algebra in order to bring you up to the level needed for your required or desired M and Q classes.
1. Always read math problems completely before beginning any calculations. If you "glance" too quickly at a problem, you may misunderstand what really needs to be done to complete the problem.
2. Whenever possible, draw a diagram. Even though you may be able to visualize the situation mentally, a hand drawn diagram will allow you to label the picture, to add auxiliary lines, and to view the situation from different perspectives.
The mission of the Wittenberg University Math Workshop is to provide support for students in mastering the mathematical tools and concepts necessary for them to attain the general education learning goal for mathematics as well as the mathematical requirements of their individual educational programs. The workshop strives to teach students to value math, to become confident in their ability to do math, to become mathematical problem solvers, and to learn to communicate and reason mathematically.
The workshop carries out its mission through the following activities:
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Intended for all students in their freshman year of college (with sufficient knowledge in high-school mathematics), this book is organized around eight fundamental mathematical processes: conjecture, logical argumentation, formal demonstration, algorithmic thinking, correspondence, enumeration, limiting processes, and approximation. Topically, the book cuts across several traditional branches of mathematics, including algebra, trigonometry, number theory, and analysis. Both formal demonstrations and problem solving with extensive applications to the physical sciences are stressed throughout. Use of the microcomputer as a working tool is also emphasized throughout the book.
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Appendices
We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to solve any system by a well-described method.
Consistent Systems
The computer scientist Donald Knuth said, "Science is what we understand well enough to explain to a computer. Art is everything else." In this section we'll remove solving systems of equations from the realm of art, and into the realm of science. We begin with a definition.
Definition CS (Consistent System)
A system of linear equations is consistent if it has at least one solution. Otherwise, the system is called inconsistent.
We will want to first recognize when a system is inconsistent or consistent, and in the case of consistent systems we will be able to further refine the types of solutions possible. We will do this by analyzing the reduced row-echelon form of a matrix, using the value of $r$, and the sets of column indices, $D$ and $F$, first defined back in Definition RREF.
Use of the notation for the elements of $D$ and $F$ can be a bit confusing, since we have subscripted variables that are in turn equal to integers used to index the matrix. However, many questions about matrices and systems of equations can be answered once we know $r$, $D$ and $F$. The choice of the letters $D$ and $F$ refer to our upcoming definition of dependent and free variables (Definition IDV). An example will help us begin to get comfortable with this aspect of reduced row-echelon form.
The number $r$ is the single most important piece of information we can get from the reduced row-echelon form of a matrix. It is defined as the number of nonzero rows, but since each nonzero row has a leading 1, it is also the number of leading 1's present. For each leading 1, we have a pivot column, so $r$ is also the number of pivot columns. Repeating ourselves, $r$ is the number of nonzero rows, the number of leading 1's and the number of pivot columns. Across different situations, each of these interpretations of the meaning of $r$ will be useful.
Before proving some theorems about the possibilities for solution sets to systems of equations, let's analyze one particular system with an infinite solution set very carefully as an example. We'll use this technique frequently, and shortly we'll refine it slightly.
Archetypes I and J are both fairly large for doing computations by hand (though not impossibly large). Their properties are very similar, so we will frequently analyze the situation in Archetype I, and leave you the joy of analyzing Archetype J yourself. So work through Archetype I with the text, by hand and/or with a computer, and then tackle Archetype J yourself (and check your results with those listed). Notice too that the archetypes describing systems of equations each lists the values of $r$, $D$ and $F$. Here we go...
Using the reduced row-echelon form of the augmented matrix of a system of equations to determine the nature of the solution set of the system is a very key idea. So let's look at one more example like the last one. But first a definition, and then the example. We mix our metaphors a bit when we call variables free versus dependent. Maybe we should call dependent variables "enslaved"?
Definition IDV (Independent and Dependent Variables)
Suppose $A$ is the augmented matrix of a consistent system of linear equations and $B$ is a row-equivalent matrix in reduced row-echelon form. Suppose $j$ is the index of a column of $B$ that contains the leading 1 for some row (i.e. column $j$ is a pivot column). Then the variable $x_j$ is dependent. A variable that is not dependent is called independent or free.
If you studied this definition carefully, you might wonder what to do if the system has $n$ variables and column $n+1$ is a pivot column? We will see shortly, by Theorem RCLS, that this never happens for a consistent system.
Sets are an important part of algebra, and we've seen a few already. Being comfortable with sets is important for understanding and writing proofs. If you haven't already, pay a visit now to Section SET:Sets.
We can now use the values of $m$, $n$, $r$, and the independent and dependent variables to categorize the solution sets for linear systems through a sequence of theorems.
First we have an important theorem that explores the distinction between consistent and inconsistent linear systems.
Theorem RCLS (Recognizing Consistency of a Linear System)
Suppose $A$ is the augmented matrix of a system of linear equations with $n$ variables. Suppose also that $B$ is a row-equivalent matrix in reduced row-echelon form with $r$ nonzero rows. Then the system of equations is inconsistent if and only if the leading 1 of row $r$ is located in column $n+1$ of $B$.
The beauty of this theorem being an equivalence is that we can unequivocally test to see if a system is consistent or inconsistent by looking at just a single entry of the reduced row-echelon form matrix. We could program a computer to do it!
Notice that for a consistent system the row-reduced augmented matrix has $n+1\in F$, so the largest element of $F$ does not refer to a variable. Also, for an inconsistent system, $n+1\in D$, and it then does not make much sense to discuss whether or not variables are free or dependent since there is no solution. Take a look back at Definition IDV and see why we did not need to consider the possibility of referencing $x_{n+1}$ as a dependent variable.
With the characterization of Theorem RCLS, we can explore the relationships between $r$ and $n$ in light of the consistency of a system of equations. First, a situation where we can quickly conclude the inconsistency of a system.
Theorem ISRN (Inconsistent Systems, $r$ and $n$)
Suppose $A$ is the augmented matrix of a system of linear equations in $n$ variables. Suppose also that $B$ is a row-equivalent matrix in reduced row-echelon form with $r$ rows that are not completely zeros. If $r=n+1$, then the system of equations is inconsistent.
Next, if a system is consistent, we can distinguish between a unique solution and infinitely many solutions, and furthermore, we recognize that these are the only two possibilities.
Theorem CSRN (Consistent Systems, $r$ and $n$ zero rows. Then $r\leq n$. If $r=n$, then the system has a unique solution, and if $r < n$, then the system has infinitely many solutions.
Free Variables
The next theorem simply states a conclusion from the final paragraph of the previous proof, allowing us to state explicitly the number of free variables for a consistent system.
Theorem FVCS (Free Variables for Consistent Systems completely zeros. Then the solution set can be described with $n-r$ free variables.
We have accomplished a lot so far, but our main goal has been the following theorem, which is now very simple to prove. The proof is so simple that we ought to call it a corollary, but the result is important enough that it deserves to be called a theorem. (See technique LC.) Notice that this theorem was presaged first by Example TTS and further foreshadowed by other examples.
Theorem PSSLS (Possible Solution Sets for Linear Systems)
A system of linear equations has no solutions, a unique solution or infinitely many solutions.
Here is a diagram that consolidates several of our theorems from this section, and which is of practical use when you analyze systems of equations.
Decision Tree for Solving Linear Systems
We have one more theorem to round out our set of tools for determining solution sets to systems of linear equations.
Theorem CMVEI (Consistent, More Variables than Equations, Infinite solutions)
Suppose a consistent system of linear equations has $m$ equations in $n$ variables. If $n>m$, then the system has infinitely many solutions.
Notice that to use this theorem we need only know that the system is consistent, together with the values of $m$ and $n$. We do not necessarily have to compute a row-equivalent reduced row-echelon form matrix, even though we discussed such a matrix in the proof. This is the substance of the following example.
These theorems give us the procedures and implications that allow us to completely solve any system of linear equations. The main computational tool is using row operations to convert an augmented matrix into reduced row-echelon form. Here's a broad outline of how we would instruct a computer to solve a system of linear equations.
Represent a system of linear equations by an augmented matrix (an array is the appropriate data structure in most computer languages).
Convert the matrix to a row-equivalent matrix in reduced row-echelon form using the procedure from the proof of Theorem REMEF.
Determine $r$ and locate the leading 1 of row $r$. If it is in column $n+1$, output the statement that the system is inconsistent and halt.
With the leading 1 of row $r$ not in column $n+1$, there are two possibilities:
$r=n$ and the solution is unique. It can be read off directly from the entries in rows 1 through $n$ of column $n+1$.
$r < n$ and there are infinitely many solutions. If only a single solution is needed, set all the free variables to zero and read off the dependent variable values from column $n+1$, as in the second half of the proof of Theorem RCLS. If the entire solution set is required, figure out some nice compact way to describe it, since your finite computer is not big enough to hold all the solutions (we'll have such a way soon).
The above makes it all sound a bit simpler than it really is. In practice, row operations employ division (usually to get a leading entry of a row to convert to a leading 1) and that will introduce round-off errors. Entries that should be zero sometimes end up being very, very small nonzero entries, or small entries lead to overflow errors when used as divisors. A variety of strategies can be employed to minimize these sorts of errors, and this is one of the main topics in the important subject known as numerical linear algebra.
In this section we've gained a foolproof procedure for solving any system of linear equations, no matter how many equations or variables. We also have a handful of theorems that allow us to determine partial information about a solution set without actually constructing the whole set itself. Donald Knuth would be proud.
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A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than focusing strictly on historical and mathematical issues — and features several helpful appendixes
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Physics Equations, Physical Quantities and Maths
A series of word documents that lists all of the equations and physical quantities encountered in the Standard Grade Physics curriculum and also a worksheet designed to help pupils cope with the maths used in Physics.
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Algebra
Homework Guidelines for Mathematics Mathematics is a language, and as such it has standards of writing which should be observed. In a writing class, one must respect the rules of grammar and punctuation, one must write in organized paragraphs built with complete sentences, and the final draft must be a neat paper with a title. Similarly, there are certain standards for mathematics assignments. You should use your instructor or grader as a study aid, in addition to the text, study guides, study groups, and tutoring services. Your work is much easier to grade when you have made your work and reasoning clear, and any difficulties you have in completing the assignment can be better explained by the grader.
Specializing in saltwater aquariums, Nic Tiemens and Joe Pineda love the challenge of recreating a slice of the ocean indoors. Day in and day out, they use volume calculations, temperature, measurement and science to create these beautiful habitats. Running time 5:25 minutes. Columbia Sportswear Designer Chris Araujo combines innovation with design to create backpacks for one of the largest outdoor apparel companies in the world.
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Maths
This qualification in mathematics encourages students to develop confidence and to develop a positive attitude towards mathematics.
It encourages students to recognise the importance of mathematics in their own lives and in society. The course is a 'Linear' model ensuring students develop their mathematical skills over two years before taking an examination. Students have the opportunity to develop, acquire and use problem-solving strategies. They will be able to apply mathematical techniques in every day and real-world situations, reason mathematically, make deductions and inferences and draw conclusions using mathematical solutions.
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Algebra: Form and Function
9780471707080
ISBN:
0471707082
Edition: 1 Pub Date: 2009 Publisher: Wiley
Summary: This text offers a fresh approach to algebra that focuses on teaching readers how to truly understand the principles, rather than viewing them merely as tools for other forms of mathematics. It relies on a storyline to form the backbone of the chapters and make the material more engaging
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Buy now
E-Books are also available on all known E-Book shops.
Short description The book describes the fundamental principles of computer arithmetic. Algorithms for performing operations like addition, subtraction, multiplication, and division in digit computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.
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Courses
COURSES FOR CREDIT
ALGEBRA I
Grades 9–12
July 8–August 9, 2013
9:30 a.m.–1:30 p.m.
$1,250
Algebra I is an introductory course that places an emphasis on the systematic development of the language through which most of mathematics is communicated. It provides the foundations for a systematic way to represent mathematical relationships and analyze change, as well as the mathematical understanding to operate with concepts at an abstract level, and then apply those concepts in a process that fosters generalizations and insights beyond the original content. Topics covered are: properties of the number system, linear functions, inequalities, operations on real numbers and polynomials, exponents, radicals, and quadratics. Successful completion of this course prepares students for Geometry and Algebra II.
ALGEBRA II
The study of functions and an extension of the concepts of Algebra I and many of the concepts of geometry are covered. Topics covered are: linear and quadratic equations and functions; systems of equations and inequalities; polynomials and rational polynomial expressions; polynomial functions; conic sections; exponential and logarithmic functions; probability and statistics. Prerequisite: Algebra I with a grade of C or better, or teacher recommendation.
SUMMER PACKETS
MATH PACKET
Grades 6–12 (C/A Students only)
August 12–16, 2013
9:30 a.m.–12:00 p.m.
$300
Open to Commonwealth Academy students only, this course is designed to help students complete their summer math packets.
READING PACKET
Grades 6–12 (C/A Students only)
August 12–16, 2013
12:30 p.m.–3:00 p.m.
$300
Open to Commonwealth Academy students only, this course is for those students in need of assistance with their summer reading work (projects, reading logs). Both books must be read prior to attending for the student to receive the full benefit of the course.
EXTENDED CARE
BEFORE AND AFTER CARE
Students enrolled in Algebra I
July 8–August 9, 2013 (Mon.–Thurs. only)
8:30 a.m.–9:30 a.m.
1:30 p.m.–3:00 p.m.
$10/hour or $20/day
We will hold before and after care services throughout the week to allow parents to drop their students off early or pick them up later in the day. The service is provided beginning at 8:30 a.m. and ends at 3:00 p.m. Payment for these services must be made in advance.
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Computational Mathematics Colleges
A program that focuses on the application of mathematics to the theory, architecture, and design of computers, computational techniques, and algorithms. Includes instruction in computer theory,cybernetics, numerical analysis, algorithm development, binary structures, combinatorics, advanced statistics, and
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Algebra Homework Help
Ordinary algebra is a topic almost everyone studies to some extent in high school. Even so, it's easy to forget basic skills, and many people find themselves having difficulty in math classes later in life because of those forgotten skills. We offer Algebra homework help to get you caught up and ready to take this subject by storm.
Typical topics in a basic, college-level class in ordinary algebra will include:
Graphs, Functions, and Models
Functions, Equations, and Inequalities
Polynomial and Rational Functions
Exponential and Logarithmic Functions
Systems of Equations and Matrices
Conic Sections
Sequences, Series, and Probability
A truly great website for getting help and extra practice in ordinary algebra at all levels is the Virtual Math Lab of West Texas A&M University.
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Math-Assist - Bernd Schultheiss
Math shareware from Germany (English, French, and German versions are available) for secondary level or high school students and teachers. Math-Assist helps in solving most of the tasks of algebra, geometry, analysis, stochastics, and linear algebra.
...more>>
Mathematica Courseware Catalog - Wolfram Research, Inc.
Developed as a resource for the academic community, the Mathematica Courseware Catalog is an ever-growing and broad-ranging collection of course materials that make significant use of Mathematica. Search by keyword or browse by subject: see, in particular,
...more>>
Mathematical Sciences Institute - Andrew Talmadge
Technology-driven professional development for middle and secondary level mathematics teachers. Week-long summer courses led by Nils Ahbel, Karen Bryant, Doug Kuhlmann, Ron Lancaster, Ira Nirenberg, and others have included Using the iPad to Enrich and
...more>>
Mathematics
This blog of instructional posts, which dates back to January, 2011, has included articles that introduce concepts of scale factor, logarithms, calculus, surds, and more.
...more>>
Mathematics and Multimedia - Guillermo Bautista
Blog about mathematics, teaching, learning, and technology. Begun in October, 2009, it has included posts such as "GeoGebra Tutorial 4 - Graphs and Sliders"; "Screencasting Tutorial: Making a Math Video Lesson Using Camstudio"; "Derivative in Real Life
...more>>
Mathematics by Mr. P - Mike Poliquin
Middle school math teacher Poliquin began this blog, subtitled "making sure everyone gets it," in December, 2011. Posts have included various original problems of the week; "Eudoxus of Cnidus: the First Mathematical Superstar"; "A Quick Dip in the Deep
...more>>
Mathematics - Student Helpmate - Chris Divyak
Search or browse this archive of questions about algebra, calculus, geometry, statistics, trigonometry, and other college math; then pay for access to answers. To submit your own problem to Student Helpmate, type your question or upload it as a file;
...more>>
Math Homework Help
Email contact for homework help in pre-algebra, algebra I and II, college algebra, geometry, trigonometry, pre-calculus, and calculus. Site also contains a math history timeline; math dictionary; some basic differentiation and integration rules; trigonometric
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Math I Can Do - P. Karl Halton
An online interactive equation editor for writing math expressions. Register a free account to save up to ten files online; purchase a premium account for unlimited storage, file sharing, and more features. Watch video tutorials of the what-you-see-is-what-you-get Motivation - Michael Sakowski
Answers to the question "Where will I ever use algebra?" Examples of how "the process of learning higher mathematics provides valuable skills in deductive reasoning and symbolic reasoning, in addition to math skills used directly in science and engineering
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Maths Is Fun - Rod Pierce, mathsisfun.com
Math revision (review) pages, games, puzzles, and offline activities for age 11 through college algebra. A discussion forum and a newsletter are also available. The Illustrated Math Dictionary links definitions to further resources on the site. Maths
...more>>
Mathsman - Mark Longson
Questions with answers and explanations, on a variety of topics. Starts with the foundations of the topic and progresses to a mastery quiz. Small fee charged for the access password.
...more>>
Maths - Martin John Baker, EuclideanSpace
Originally intended to give enough maths information to allow physical objects to be simulated by a computer program, these pages now cover a broader range of mathematical topics. The pages that get the most hits on the site are those concerned with 3D
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Pages
Maths
Mr Robb's Math Videos is a YouTube channel containing 555 videos produced by high school mathematics teacher Bradley Robb. Mr. Robb's videos explain and demonstrate solving problems in Algebra I, Algebra II, and Calculus. Most of the videos are recorded while Mr. Robb is teaching
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Textbook Experiencing Geometry: Euclidean and
Non-Euclidean with History (Third Edition), David W. Henderson and
Daina Taimina
Purposes
To develop a deep and personal understanding of
Euclidean, spherical and hyperbolic geometries and how they relate to
measuring the universe in which we live.
Overview
This course will take a more philosophical perspective on
geometry rather than a computational or result-based perspective. In
this class we will use several different methods to analyse
geometries. We will rarely have traditional lectures. More
frequently will be times for individual work, group work, and class
discussion. While there will be some familiar looking homework
exercises, there will mostly be more personal discussions of homework
problems.
Reading
I have intentionally chosen Henderson'€™s book as an
exploratory and philosophical text. In addition to planning time to do
homework, please take time to carefully read the chapters in the book.
Notice use of the words '€œtime'€ and '€œcarefully'€. Read the
sections slowly. Read actively, that is while writing and with models
at hand. If you do not understand some statement reread it, think of
some potential meanings and see if they are consistent, and if all else
fails, ask me. If you do not believe a statement, check it with your
own examples. Finally, if you understand and believe the statements,
consider how you would convince someone else that they are true, in other
words, how would you prove them?
Learning Outcomes
Upon successful completion of Math 335 a student will be
able to
'€Ę Compare and contrast the geometries of
the Euclidean and hyperbolic planes.
'€Ę Analyze axioms for the Euclidean
and hyperbolic planes and their consequences.
'€Ę Use transformational and axiomatic
techniques to prove theorems.
'€Ę Analyze the different consequences
and meanings of parallelism on the Euclidean and hyperbolic planes.
'€Ę Demonstrate knowledge of the
historical development of Euclidean and non-Euclidean geometries.
Grading
Your grade in this course will be based upon your
performance on two styles of homework, a project, an in-class exam and a
final experience. The weight assigned to each is designated below:
Homework Problems 3/5
Project
1/5
Final Experience
1/5
Homework Problems
Throughout the course you will write up discussions of
'€œProblems'€ from Henderson'€™s book. Before these papers are handed
in, I strongly suggest somehow submitting drafts to me for comments.
These drafts are not required, but will strengthen your understanding and
your final products. Drafts can either be submitted in paper or via
email. Either way I will return them with comments and
suggestions. The end goal of writing each problem will be presenting
your complete understanding of the question in a well-written
discussion. These discussions will be graded on a ten point decile
scale based on completeness, accuracy and writing.
These problems will be evaluated similarly to evaluating
papers in an English class.
0 missing or plagiarised
3 question copied, nothing written
6 something written that appears that it was only written
to take up space
7 substantially incomplete. Something written, but
does not really answer the main questions. Major errors.
Very poor writing
8 mostly complete. maybe a few minor errors
9 complete, no errors, some personal insight,
well-written
10 wonderful (includes concise, and to the the
point directly)
Solutions and Plagiarism
There are plenty of places that one can find all kinds of
solutions to problems in this class. Reading them and not referencing
them in your work is plagiarism, and will be reported as an academic
integrity violation. Reading them and referencing them is not quite
plagiarism, but does undermine the intent of the problems. Therefore,
if you reference solutions you will receive 0 points, but you will *not* be
reported for an academic integrity. Simply 'ۦ please do not read any
solutions for problems in this class.
Projects
Each student is responsible for completing a project as
part of a pair. A project will consist of reading one of chapters
11,14-22 from Henderson'€™s book or a similar portion of another book not
discussed in class (I have several options you may examine). The
materials for the projects must be chosen by February 22. Each project
will include a write-up of all the problems in the chosen part.
Finally, each of the projects will be presented in the last two weeks of
class.
Final Experience
The final experience will include extensive writing and
focused on summarizing the experience of different aspects of the
course. This product will be due at the time of the scheduled
final exam, May 12, 12-3p, when we will also meet to discuss the topic and
the course as a whole.
Geometer'€™s Sketchpad
We may occasionally using Geometer'€™s Sketchpad as a
method of gaining intuition for geometry. Details for working with
this software will be described in class no later than February 4 of
plans to observe the holiday.
Schedule
January 25 '€" February 13 Discuss Chapters
1-5 of Henderson
February 4
As a homework exercise,
show a model of a hyperbolic plane
February 18 - March 10 Discuss
Chapters 6, 7, 9 of Henderson
February 22
Final write
up of Chapters 1-5 '€œProblems'€ due
February 22
Projects
must be chosen by this date.
March 11 - 29
Discuss Chapters
8, 10 of Henderson
March 15
Drafts of Chapters 6, 7, 9 will not be accepted after
March 29
Final write up of Chapters 6, 7, 9 '€œProblems'€ due
April 1 '€" 18
Discuss Chapters 11, 12 of
Henderson, and Constructions
April 5
Drafts of Chapters 8, 10 will not be
accepted after
April 12
Final write up of Chapters 8, 10 '€œProblems'€ due
April 21
Drafts of Chapters 12, 13 will not be accepted
after
April 26 '€" May 6
Project presentations
May 6
Written projects due. Final write up of
Chapters 12, 13 "Problems" due
Wednesday May 15, 12-3p Final
experience - leftover presentations.
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Get ready to master the principles and operations of algebra! Master Math: Algebra is a comprehensive reference guide that explains and clarifies algebraic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced topics that will help prepare you for pre-calculus... more...
This volume consists of a collection of survey articles by invited speakers and original articles refereed by world experts that was presented at the fifth China–Japan–Korea International Symposium. The survey articles provide some ideas of the application as well as an excellent overview of the various areas in ring theory. The original... more...
Karoubi 's classic K-Theory, An Introduction is to provide advanced students and mathematicians in other fields with the fundamental material in this subject . K-Theory, An Introduction is a phenomenally attractive book: a fantastic introduction and then some. serve as a fundamental reference and source of instruction for outsiders who would be... more...
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a practical guide for teachers to help students develop thinking skills in the areas of geometric properties, transformations, and measurement of geometric objects; and includes advd that features students working through open-ended problems.
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Share and embed TI-Nspire documents online: create homework for the class or lesson ideas for colleagues. Create a new Web page in a few simple steps or add playable TI-Nspire documents to your Web page or blog – ideal for sharing your ideas with users who don't yet have TI-Nspire software installed.
TI-Nspire Student Software provides you with the same functionality as the handheld: Calculator, Graphs, Geometry, Lists & Spreadsheet, Data & Statistics, Notes, Programming and Vernier DataQuest™. Download the full software as either a .exe or zip file for Windows, or a Mac App. Fully functional for 30 days. Buy a permanent licence key below.
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Fort Worth Precalculus both steps are meaningless if you don't know or don't understand what's going on. You need to understand what you're doing and why. In another word, what you do should make sense to you
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Which topics/problems could you show to a bright first year mathematics student?
I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other management or business sciences etc., the course has to be a generic one. By this I mean that we teach Calculus almost exclusively. Sure, there are topics like the Binomial Theorem and general remarks of proving theorems, but students that are interested in mathematics don't find the material particularly interesting. What is more, I can relate to them since I found the first two years of university mathematics somewhat boring. This included calculus, linear algebra, convergent/divergent sequences, multiple integrals etc. Real analysis (except for sequences), complex analysis, abstract algebra, topology and even elementary number theory do not appear until the third year.
The sad thing is that many students take mathematics only up to second year and do not get to see any of the "cool"/"interesting" mathematics even though they might be interested in mathematics. So I wondered: Is there any way of introducing "interesting" mathematics to them? ("Them" - in particular first years, but this question is also relevant for second years, who might have a higher degree of maturity.)
Some things that I (and the other lecturers of this course) have thought of are:
Adding challenging questions to tutorials (e.g. IMC or Putnam, though these are harder than we would like)
Writing short introductory "articles" about fields or groups or perhaps the Euler characteristic (as an introduction to topology) etc. Of course, this is very idealistic, since one often doesn't have time or energy to do this.
Referring them to library books where some of these things are explained. Also, quite idealistic, but how many will actually go to the library.
The best solution is probably to combine the three. Have a question which has a strange answer or solution, which can be explained by some interesting mathematics. Shortly explain how this is done, and have a reference where the student can go if he is interested enough to pursue it further.
Are there any other ways of achieving this goal? Do you know of any questions to which this (combined) procedure can be applied?
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This "book" isn't an explanation of mathematics at all; sure, I can accept that. My biggest qualm is that it isn't even written in Pushtu; sure, it has the same letters. Sure, the vocabulary could be bloody freaking accurate, but this book is written with English grammar, AND from LEFT to RIGHT! (This is a right to left language BTW.) Its like trying to read something below.
.is stupid above book The (the book above is stupid) ~This is written with Pushtu grammar BTW. .think is it I (I think it is)
Get the picture? Its an absolute waste of time, effort and money. Go elsewhere and don't even spend time considering this worthless book. There wasn't any effort put into it despite the notion otherwise.
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This is a free, online textbook offered by American Mathematical Society (AMS). "The sescond semicentennial volume contains brief treatises on eight representative subjects and a historical summary of American contributions of mathematics during the past fifty years."
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Primary Mathematics
9781442505636
Pages: 600 Publication Date: 19 November 2009 Format: Paperback Availability: Available
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you.
Online Price $109.10RRP $123.95Save $14 fourth edition has been significantly revised and updated for the current educational environment and has been written with a greater emphasis on how students learn mathematics. It provides teachers and students with a sound framework for the successful teaching of mathematics to primary students. It is suitable both as a core text for primary student teachers and as an indispensable reference for practising primary teachers seeking to update their knowledge.
9781442505636
ISBN 10: 144250563 Pages: 600 Publication Date: 19 November 2009
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Math 0306 Sections
1.1 Introduction to Whole Numbers 1.2 Addition and Subtraction of Whole Numbers 1.3 Multiplication and Division of Whole Numbers 1.4 The Order of Operations Agreement
2.1 LCM and GCF 2.2 Introduction to Fractions 2.3 Addition and Subtraction of Fractions 2.4 Multiplication and Division of Fractions 2.5 Introduction to Decimals 2.6 Operations on Decimals 2.7 The Order of Operations Agreement
3.1 Introduction to Integers 3.2 Addition and Subtraction of Integers 3.3 Multiplication and Division of Integers 3.4 Operations with Rational Numbers 3.5 The Order of Operations Agreement
12.1 The Rectangular Coordinate System 12.2 Linear Equations in Two Variables 12.3 Intercepts and Slopes of Straight Lines 12.4 Equations of a Straight Line
13.1 Solving System of Linear Equations by Graphing 13.2 Solving System of Linear Equations by the Substitution Method 13.3 Solving System of Linear Equations by the Addition Method 13.4 Application Problems in Two Variables
MATH 2413 - Calculus I, 4 Credits
Textbook
Math 2413 Sections
2.1 The Tangent and Velocity Problems 2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.4 The Precise Definition of the Limit 2.5 Continuity 2.6 Limits at Infinity 2.7 Derivatives and Rates of Change 2.8 The Derivative as a Function
3.1 Derivatives of Polynomials and Exponential Functions 3.2 The Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 The Chain Rule 3.5 Implicit Derivatives 3.6 Derivatives of Logarithmic Functions 3.7 Rates of Change in the Natural and Social Sciences
11.1 Sequences 11.2 Series 11.3 The Integral Test and Estimates of Sums 11.4 The Comparison Tests 11.5 Alternating Series 11.6 Absolute Convergence and the Ratio and Root Tests 11.7 Strategy for Testing Series 11.8 Power Series 11.9 Representations of functions as Power Series 11.10 Taylor and Maclaurin Series 11.11 Applications of Taylor Polynomials
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The issue Jacques describes has nothing to do with watering down of standards. Four semesters of math is (or should be) sufficient to get through upper-division Quantum Mechanics. Linear Algebra issues aside, the difficulty of upper-division Quantum Mechanics stems from the conceptual issues, not the math. The problem, as Jacques points out, is a mismatch between the math curriculum and the physics curriculum.
The UT physics department could do one of three things to fix this problem.
Require all physics students to take, or test out of, both the Advanced Calculus class and the Linear Algebra class. Physics majors really ought to have both under their belts.
Forget about those two a la carte classes. Instead, require all upper-division majors to take a "Mathematical Methods for Physicists" class, designed to ensure that everyone has the right machinery to forge through their upper-division work.
Coordinate with the math department; adjust the mathematical core accordingly.
My alma mater used the third approach. The math, engineering, computer science, and natural science departments all coordinated closely on the base four-semester mathematics core. Individual departments could then layer additional requirements, but at least everyone had a common foundation, even the biologists and computer scientists. This solution worked great for a school with 700 undergrads, where all the professors knew each other personally, shared babysitters, and so on. It would probably work less well for UT.[1]
1. The main disadvantage of this "Grand Unified Core" approach is that it generates a great deal of whining from certain students over "taking math that I'll never use!" Long ago, I used to sympathize with my oppressed computer science and biology brethren. But now... not so much. Over the last few years, I have run into senior developers who did not understand that ln (A + B) is not equal to ln A + ln B. And who when queried about this responded, "Look, I have a mathematical background, I really can't explain it to you." Professors of All and Sundry Technical Disciplines: please don't let this happen to your graduating seniors. Thank you.
Comments
Every once in a while, I regret the fact that I didn't get to take any math classes in my undergraduate days. …the last time was in Fall 1999….and I was able to get by with trigonometry.
I could never figure out why all the lower division math classes at my university met 5 days a week at 8am. There was no way to take math and sleep late! I felt cheated.
It made no sense because all lower division language classes met the same time…that's why I don't speak 3 languages and possiblly why I had to write it out the stuff about 1nA + 1nB to see what Evan meant.
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Mathematics
The goal of the Middle Creek High School mathematics program is to provide high-quality instruction to enable students to solve problems creatively and resourcefully, to compute fluently, and to prepare them to fulfill personal ambitions and career goals in an ever-changing world.
The Middle Creek High School chapter of Mu Alpha Theta (Math Honor Society) is an organization whose purpose is to promote scholarship in, and enjoyment and understanding of, mathematics among high school students. We do this in a variety of ways including math tutoring, competitions, guest speakers and meetings. We usually meet on the first Monday of every month. Because this is an honor society, there are certain requirements for membership (see below), but we welcome anyone who wants to learn more about math to our activities. Meeting dates and other information is located on Mrs. Katie Taylor's blackboard site.
Membership information: Full members are students who have completed four semesters math starting with Algebra 1 and are (at least) enrolled in a fifth semester of math with at least a 3.0 (unweighted) GPA for these math courses. The one time membership dues are $16, which includes a graduation cord provided to members who are still in good standing at the end of their senior year. Full members are required to accumulate 25 gold stars throughout the year and may be attained by attending meetings, tutoring, competitions, and gold star problems.
Associate members are students who have completed two semesters of math starting with Algebra 1 and are enrolled in a third math with at least a 3.0 (unweighted) GPA for these math courses. They are expected to participate in the meetings and may also tutor, participate in the competitions and submit gold star solutions.
Members of the Math Honor Society also host a tutoring center every Tuesday and Wednesday from 2:30-3:30 in Mrs. Taylor's room #8407. This is for ALL math classes.
Below is a list of Math Tutors that has been generated by the WCPSS Math Office. Families may access the list if they are interesting in tutoring options outside of what Middle Creek HS Math teachers offer at school.
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Math 300: Mathematical Computing
Math 300 Syllabus
Welcome to Math 300 - Mathematical Computation. The goal of this course
is to make you more sophisticated in your knowledge of computing in
mathematics. Anyone can use a browser and a word processor, but mathematicians
and teachers need an array of more specialized techniques to do and communicate
mathematics in myriad formats. Mathematicians need unique powerful tools
to analyze their problems, and use multiple platforms for those ends.
To that end, we try to familiarize you with some of the most common
aspects of operating systems, networking, typesetting, and applications
that mathematicians use.
Instructor:
Kevin Cooper
Office:
Neill 322
Office Hours:
3-5 MWF: these will ordinarily be in room 120
Phone:
5-4771
Email:
kcooper@math.wsu.edu
Tests:
There will be two tests worth a total of 200 points.
Assignments:
There will be several assignments worth about 400 points. These will
typically involve solving a problem and writing about the solution,
and then typesetting that writing in some way. Several of the assignments
in this course include substantial writing components. You will be graded
on writing as well as computational understanding. Thus, technical proficiency
alone will not suffice to do well in the class.
Text:
This is it. There are some HTML text pages available at this site,
as well as somewhat more complete notes in portable document format.
There are other resources available on the Web to which we provide links.
Academic Integrity:
Because much of the work in this class is done electronically, some
students find it too tempting to copy the work of others. While we encourage
collaboration and helpfulness among students, ultimately students must
demonstrate that they have learned something by turning in their own
work. Assignments or exams that show clear evidence of plagiarism (copying)
will receive scores of zero, or in egregious cases might
lead to a failing grade in the class.
This can apply regardless of whether the
student in question was the one copying, or the one copied. Protect
your own work.
Topics:
Working Remotely
Operating Systems - especially Unix
HTML, MathML, XML
Tex and Latex2html - document formatting
Python - including programming
Students with disabilities
Reasonable accommodations are available for students with a documented disability. If you have a disability and need accommodations to fully participate in this class, please either visit or call the Access Center (Washington Building 217; 509-335-3417) to schedule an appointment with an Access Advisor. All accommodations MUST be approved through the Access Center.
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Book Description: This workbook's fifth edition has been updated to reflect questions and question types appearing on the most recent tests. Hundreds of math questions in both multiple-choice and grid-in formats with worked out solutions Math strategies to help test-takers approach and correctly answer question types that might be unfamiliar to them All questions answered and explained Here is intensive preparation for the SAT's all-important Math section, and a valuable learning tool for college-bound students who need extra help in math and feel the need to raise their math scores.
Buyback (Sell directly to one of these merchants and get cash immediately)
Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it
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textbook on linear algebra includes the key topics of the subject that most advanced undergraduates need to learn before entering graduate school. All the usual topics, such as complex vector spaces, complex inner products, the Spectral theorem for normal operators, dual spaces, the minimal polynomial, the Jordan canonical form, and the rational canonical form, are covered, along with a chapter on determinants at the end of the book. In addition, there is material throughout the text on linear differential equations and how it integrates with all of the important concepts in linear algebra. This book has several distinguishing features that set it apart from other linear algebra texts. For example: Gaussian elimination is used as the key tool in getting at eigenvalues; it takes an essentially determinant-free approach to linear algebra; and systems of linear differential equations are used as frequent motivation for the reader. Another motivating aspect of the book is the excellent and engaging exercises that abound in this text. This textbook is written for an upper-division undergraduate course on Linear Algebra.The prerequisites for this book are a familiarity with basic matrix algebra and elementary calculus, although any student who is willing to think abstractly should not have too much difficulty in understanding this text. less
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Other Course Material: A graphing calculator , Texas Instruments TI-83, TI-83
Plus, TI-84, or TI-
84 Plus is required for this course. Each student is expected to have access to
his or her own
calculator. Calculators may be used for all work but may not be shared during
quizzes, tests
and/or examinations. Calculators with symbolic manipulations capabilities are
not allowed on
quizzes, tests, and/or examinations. Some examples of these are TI-89, TI-92,
and HP-48 series.
LEARNING OUTCOMES AND EVALUATION PROCEDURES
KNOWLEDGE:
After completing this course, the student will:
1. Know the concepts of a function, the graphs of functions, inverse functions,
composition of
functions, and other properties of functions .
2 Identify polynomial and rational functions.
3. List and cite the zeros of polynomial and rational functions.
4. Know the various graphing and algebraic approaches to solving systems of
linear equations
and inequalities.
5. Know how to use a graphing calculator and a computer software tutorial.
6. Relate reading and writing skills to describe mathematical concepts and
critical thinking
processes to the solutions of various word problems.
7. Know graphic, symbolic, and verbal re presentations of problems .
8. Define exponential and logarithmic functions .
9. Know how to solve exponential and logarithmic equations.
1. Identify and interpret the graphs of functions, inverse functions, and
composition of
functions.
2. Classify and apply polynomial and rational functions.
3. Find the zeros of polynomial and rational functions.
3. Use the various graphing and algebraic approaches to solve systems of
equations and
inequalities .
5. Use the graphing calculator and work through a computer software tutorial.
6. Apply reading and writing skills to describe mathematical concepts and
critical thinking
processes to the solutions of various word problems.
7. Interpret graphic, symbolic, and verbal representation of problems.
8. Recognize exponential and logarithmic functions, their properties, and their
graphs.
9. Apply various properties of exponential and logarithmic functions.
10. Demonstrate the difference between natural and common logarithms.
11. Draw exponential and logarithmic functions using a graphing calculator.
After completing this course, the student will:
1. Create and analyze the graphs of functions, inverse functions, and
composition of functions.
2. Analyze polynomial and rational functions.
3. Evaluate polynomial and rational functions.
4. Compare and test the various graphing and algebraic approaches to solve
systems of
equations and inequalities.
5. Analyze problem situations using a graphing calculator and a computer
software tutorial.
6. Integrate reading and writing skills to describe mathematical concepts and
critical thinking
processes to the solutions of various word problems.
7. Create graphic, symbolic, and verbal representation of problems.
8. Solve a system of linear equations using the elimination andsubstitution
methods .
9. Compare exponential and logarithmic functions and their characteristics,
including their
graphs.
10. Distinguish and relate the natural logarithm to the common logarithm.
11. Plot exponential and logarithmic functions using a graphing calculator.
1. A common departmental midterm examination.
2. A common departmental final examination.
3. Other work as assigned by the instructor.
COURSE REQUIREMENTS:
The student is required to:
1. Attend class regularly and on time.
2. Be present on test dates.
3. Turn in written homework assignments by the due date.
4. Purchase and read the required texts and related material.
5. Purchase and learn to effectively use a graphing calculator.
6. Participate actively in class problem solving exercises.
7. Perform satisfactorily on examinations.
The University=s policies on specific academic regulations concerning cheating,
plagiarism,
absenteeism, etc. will be adhered to in this course. These policies are stated
in the VSU
Undergraduate Catalog and the VSU Student Handbook.
Students with learning or other disabilities who are covered under the American
Disability
Act should privately inform the teacher of this fact so that appropriate
instructional
arrangements can be made.
EVALUATION STRATEGIES:
The student will be evaluated on four tests, a midterm exam, a common
departmental
comprehensive final examination, written and online homework, quizzes or other
work assigned by
the instructor.
Section 12.1 – Systems of Linear Equations:
Substitution and Elimination (Systems of two equations in two variables)
If time permits, additional sections might be covered at the discretion
of your
instructor.
The midterm exam will cover Chapters 3 and Chapter 4.
The final exam will be comprehensive and include all the sections covered in
this course
during this semester
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Is
there an abstraction of the ideas running through a number of
disparate areas of mathematics which, when accurately defined,
gives greater insight into these individual areas? Does
this abstraction allow for discoveries which apply to the whole
without consideration of the characteristics which distinguish
the parts?By
reaching the level of abstraction which Modern Algebra provides,
we'll explore some of the most beautiful and unifying results in
mathematics.
Objectives:
Master the abstraction
necessary for working with groups, rings, and fields
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Both Calculus AB and Calculus BC are covered in this comprehensive AP test preparation manual. Main features include four practice exams in Calculus AB and four more in Calculus BC with answers and solutions explained, a detailed subject review covering topics for both exams, and advice to students on efficient use of their graphing calculators.
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Questions Posted by TheExpert92
Part A Given: †
Contextual factors include community, school, and classroom factors; characteristics of students; studentsí varied approaches to learning; and studentsí skills and prior learning. It is...
Task 2 Introduction:
†
It is important to understand applications of recursion that would be useful for teaching high school mathematics.
†
Scenario:
†
You are a mathematics teacher at a high school....
task 1 Introduction:
†
Mathematical modeling is a powerful scientific tool, and knowing about it increases the competence of mathematicians and mathematics teachers.
†
Given:
†
The data listed in the...
lesson plan 2 Task C:
†
A.† Using the attached lesson plan format, create an original lesson plan to describe the historical development of Euclidean and non-Euclidean Geometries . The lesson will include:
...
lesson plan 2 Many diverse cultures have contributed to the development of mathematics over time.
†
Task:
†
†
A. † Using the attached lesson plan format, create an original lesson plan to describe the...
task 4 Fields are an important algebraic structure, and complex numbers have that structure.
Task:
A. Use de Moivre's formula to verify that the 5th roots of unity form a group under complex...
algebra 3 Introduction:
†
Rings are an important algebraic structure, and modular arithmetic has that structure.
†
Recall that for the mod m relation, the congruence class of an integer x is denoted [ x] m....
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MAT-031 Developmental Mathematics
Credits: 3 Developmental Mathematics is intended for students who need assistance in basic arithmetic skills. Based on assessment of student needs, instruction includes performing the four arithmetic operations with whole numbers, fractions, decimals, and percents. Application skills are emphasized. 3.0-3.0-3.0
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Math 125 - Intermediate Algebra with Applications
Course Description
This course is designed for the intermediate algebra student who plans to continue on to MATH 300, 310, 320, 325, STAT 300, 301, or complete an associate degree. It does not fulfill the prerequisite for MATH 315, 330, or higher numbered math courses. Topics include linear functions, models, systems, and graphs, as well as polynomial, exponential, logarithmic, and quadratic functions. The course emphasizes authentic applications and mathematical models using real-world data.
Student Learning Outcomes
Upon completion of this course, the student will be able to:
identify and solve various types of equations and systems of equations.
factor a variety of polynomials.
collect like terms in simplifying polynomial, exponential, and logarithmic functions.
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LESSON PLANS
Understanding Quadratics
Introduction
Quadratic functions are explored in various forms from Algebra 1 through college level math. These functions can be used to model the motion of objects falling to earth. If you know an object's initial velocity and height, you can find out how long it will take for it to hit the ground if it is dropped. In this activity, students will:
• Solidify their understanding of standard form and vertex form for the equation of a quadratic.
• Gain a firm understanding of the real roots of a quadratic function.
• Learn how the discriminant is calculated.
• See the relationship between the equations and graphs of quadratic functions.
• Use quadratic functions to solve free-fall problems
Materials
This is a software only lab, so you just need to have LabVIEW installed
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Math Java Applets (Popularity: ): About 15 applets covering a number of math problems and principles. Manipula Math with Java (Popularity: ): Over 200 applets for middle school students, high school students, college students, and all who are interested in mathematics. Interactive programs and a lot of animation that helps with understanding ... Java Demos for Probability and Statistics (Popularity: ): College professor's applets. Chaos and Fractals Applets (Popularity: ): Several java applets for use in exploring the topics of chaos and fractals. Experimental Math Applets (Popularity: ): Some applets covering Besicovitch sets, conformal compactifaction, honeycombs, exponent calculator, the complex plane, elementary complex maps, Möbius transforms, multi-valued functions, the complex derivative, the complex integral, Taylor and Laurent expansions. Spirograph Applet (Popularity: ): Makes a spirograph, just like the kid toy. TenBlocks and IntegerZone (Popularity: ): TenBlocks turns the times tables into a series of puzzles. IntegerZone lets users explore aspects of arithmetic and number theory using the integers themselves as the interface. Graph Explorer (Popularity: ): A Java applet for graphing functions, with smooth zooming and panning across graphs, and variable parameters which can be used for animation. Java Applets for Visualization of Statistical Concepts (Popularity: ): These applets are designed for the purpose of computer-aided education in statistic courses. The intent of these applets is to help students learn some abstract statistics concepts easier than before. ... xFunctions (Popularity: ): The xFunctions applet covers several aspects of calculus and pre-calculus mathematics, including graphs, parametric curves, derivatives, Riemann sums, and integral curves.
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Apprenticeship & Workplace Mathematics 12
Prerequisite: Apprenticeship & Workplace Mathematics 11
This course includes the following topics: purchasing vehicles, small business liability, polygons, transformations, puzzles, precision and accuracy of instruments, probability, linear relations, central tendency, sine and cosine law.
This is a course for students who will be going directly into the work force or into some trades.
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Course Communities
Trapping Lab - Linking Equations and Graphs
Course Topic(s):
Developmental Math | Systems of Equations
"This activity reinforces the relationship between the solution to a system of equations and the intersection of their corresponding graphs. Generally, students begin to solve systems by using graphing and then algebra. Once a student learns to solve the system by algebra, they often forget the connection to the graph. Hence in this lab, they will use algebra first and then graph their answers.
"
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ALGEBRA I B, the second course in a two-semester series, continues to build on students' knowledge as they learn to solve systems of linear equations and inequalities. Assessments include self-check quizzes, audio tutorials, and interactive games. Students will study units that allow them to gain practical mastery in reading, writing, and evaluating mathematical expressions. Students will study topics including polynomials, factoring, quadratic functions, and radicals. The course concludes with a study of rational expressions.
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Choose a format:
Paperback
Overview
Book Details
Differential Equations for Dummies
English
ISBN:
0470178140
EAN:
9780470178140
Category:
Mathematics / Differential Equations / General
Publisher:
Wiley & Sons, Incorporated, John
Release Date:
05/30/2008
Synopsis:
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
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INTRODUCTION
Who is this website for?
Abstractmath.org is designed for people who
are beginning the study of some part of abstract math. This includes:
University
math majors or beginning
grad students taking math courses that require working with
abstract definitions and understanding and creating proofs.
Teachers
of university courses like those just described.
Professionals who need to learn math (in any
one of many fields) that is described in terms of mathematical properties with no reference to applications.
Anyone who is curious about advanced math!
What
is abstract math?
Abstract math is my name for what is often
called "higher math" or "pure
math".
All math involves manipulating symbols (solve a
quadratic equation, find a derivative, and so on). Abstract
Math provides the conceptual background and theory
that justifies these manipulations and explains their real-world applications.
Abstract math requires conceptual
reasoning about abstract ideas (as well as manipulating symbols), in
particular on understanding and constructing proofs.
Abstract math is mathematics for its own sake. In doing abstract math, you state theorems and
prove them mostly in the context of mathematical
ideas rather than applications or ideas from other
fields.
When you first meet up with abstract math (see appendix)
you may find it hard
to understand or even bizarre. If you need to know some piece
of abstract math you may find the texts in the subject appear to be
unmotivated and
full of mysterious chains of reasoning. This happens to many
people who are quite good at solving trig, derivative and integral problems.
Overview
of the site
This website is a multiple-entry site with many cross-links. This overview
will give you a start on finding out what is on it.
The four main parts of the website
This list contains links to the head page of each of the four main parts of abstractmath.org. These head pages explain the ideas of that
part in more detail.
Many of the important ideas about mathematics in this site
are summarized in
Slogans in Purple
Prose Displayed Like This
It takes work to understand all the ins and outs
of these purple-prose slogans.
Many of them require thinking about things in a
way that is very different from the way you think about things in daily
life.
Some of them are difficult to believe and put
into practice.
Abstractmath 2.0
Some of the articles on abstractmath, including this one, are headed Abstractmath 2.0. These articles are new or new revisions of old articles using a much more efficient system of presenting math on the web. This represents a new start on abstractmath after several years of very little change. The reasons for this and the new system are discussed in my post Writing math for the web.
This new system consists of combining the use of several well-known applications and is only new to me, not to the world.
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Schaum's Outline Of Trigonometry - 3rd edition
Summary: Updated to match the emphasis in today's courses, this clear study guide focuses entirely on plane trigonometry. It summarizes the geometry properties and theorems that prove helpful for solving trigonometry problems. Also, where solving problems requires knowledge of algebra, the algebraic processes and the basic trigonometric relations are explained carefully. Hundreds of problems solved step by step speed comprehension, make important points memorable, and teach p...show moreroblem-solving skills. Many additional problems with answers help reinforce learning and let students gauge their progress as they goGoodwill BookWorks Austin, TX
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ositions, Lagrange's talks feature both originality of thought and elegance of expression. The five lectures begin with discussions of arithmetic that focus on fractions and logarithms as well as theory and applications. Subsequent talks consider algebra, with emphasis on the resolution of equations of the third and fourth degree, the resolution of numerical equations, and the employment of curves in the solution of problems. Students, teachers, and others with an interest in mathematics will find this volume a unique reading book in mathematics, with fascinating historical and philosophical remarks by a distinguished mathematician
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Tips for Success in Math 3113-3118 Come to class prepared. This means with your homework ready to turn in, prepared to discuss or present the assigned problems, and having read the next section of the text. Note that we will collect homework before
Math 3118Name_This exam is open book and open notes. Calculators are allowed, but probably won't be very helpful. Correct answers without justification will receive no credit. When you're using choose notation, please explain what is being picked
Software Engineering ICSci 5801 Summer 2008 Take Home FinalThis is a take home test. You have all the time in the world. It is an open book test and you are free to use the textbook, any material handed out in class, and any other resources. Note,
Homework-5To: CC: From: Date: Re:CSci 5801, All Students All TAs Dr. Heimdahl 7/10/2008 ASW Implementation.The ProblemWe have a design for the ASW, the customer wants it, and we need to build it.The AssignmentImplement the ASW design you han
Software Engineering ICSci 5801 Summer 2008 Take Home MidtermThis is a take home test. You have all the time in the world. It is an open book test and you are free to use the textbook, any material handed out in class, and any other resources.No
CombiMap explanationCombiMap is a transform for mapping input features, L dimensional data space, into one dimension (mapping multi-dimensional data to a scalar value). Mathematical representation of this transform has four terms that are:CombiMap
Answer's to the Tornado QuizDark or greenish skies, wall cloud, large hail, loud roar that sounds like a freight train. 2.) 3-4 days 3.) A tornado watch means there could possibly a tornado. 4.) A tornado warning means a tornado has been spotted by
AnthemClass DiscussionECO 284 Microeconomics Dr. D. Foster Is there scarcity in Anthem? How are choices made? What? How? For whom? What is the moral contrast? What sentiment is collectivism trying to usurp? How is individualism a thre
CVEN 1317: Introduction to Civil Engineering - Homework 1 [25 pts total] On a separate sheet, answer the following based on class web notes or links (http:/ceae.colorado.edu/~silverst/cven1317/). Your assignment should be typed/printed (1 point for f
Determine the Specific Heat of a Solid in a CalorimeterAREN 2110 ITL Lab AssignmentCalorimeter is a multicomponent, adiabatic process1st Law Statement: Ui = 0 Where components are the calorimeter mass and the sample mass. Assumptions: rapid heat
CVEN 5534: Wastewater Treatment Assignment 1: Due Tuesday, 1/20BACKGROUND In 1905, Pennsylvania passed a law forbidding the discharge of untreated sewage from new sewerage extensions and extensions of existing sewerage systems into streams. The law
AREN 2110: Thermodynamics Midterm 1 Fall 2005_ NameTest is open book and notes. Answer all questions and sign honor code statement: I have neither given nor received unauthorized assistance during this exam. Signed__Remember to show your work
AREN 2110: Thermodynamics Midterm 1 Fall 2004_ NameTest is open book and notes. Answer all questions and sign honor code statement: I have neither given nor received unauthorized assistance during this exam. Signed__Remember to show your work
AREN 2110: Thermodynamics Midterm 2 Fall 2005_ NameTest is open book and notes. Answer all questions and sign honor code statement: I have neither given nor received unauthorized assistance during this exam. Signed__Remember to show your work
AREN 2110: In class exercises 1st Law 1. 7.2 MJ of work is put into a gas at 1 MPa and 150 C while heat is removed at the rate of 1.5 kw. What is the change in internal energy of the gas after one hour? a. 5.7 MJ b. 1.8 MJ c. 8.7 MJ d. 13 MJ 2. One k
FCS Core Learner Outcomes1. Articulate the historical foundation of family and consumer sciences, its evolution over time, its mission, and its integrative focus. 2. Analyze family structures and apply major theoretical perspectives to understand in
Civil EngineeringWhat is Civil Engineering? What can you do as a Civil Engineer? Curriculum at CU "Engineers solve ill-defined problems that have no single "right" answer but many better or worse solutions." Engineering and the Mind's Eye, Fergus
DISCUSSION P APERBOTTLEDWATER:UNDERSTANDING A SOCIAL PHENOMENONCatherine FerrierApril 2001This report, commissioned by WWF, is an independent documentation of research by the author and its contents ultimately the responsibility of the au
DPD Portfolio Evaluation Format Includes the necessary components in the following order (12 points): Cover sheet in outer `pocket' of the binder. Title page (same as cover sheet) Table of Contents Current resumeFCS 4150Professional goals within
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SAS Programming: The One-Day Course is an introduction to using the SAS programming language. It is intended to give the reader a start in SAS programming and the basic data manipulations and statistical summaries that are available through SAS. Unlike other introductory competitors on the market, it is a pocket-sized reference that does not clutterSupermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of differential... more...
The book is devoted to universality problems. A new approach to these problems is given using some specific spaces. Since the construction of these specific spaces is set-theoretical, the given theory can be applied to different topics of Topology such as: universal mappings, dimension theory, action of groups, inverse spectra, isometrical embeddings,...Focuses on two fundamental questions in the theory of semisimple Lie groups: the geometry of Riemannian symmetric spaces and their compactifications; and branching laws for unitary representations - restricting unitary representations to subgroups and decomposing the ensuing representations into irreducibles. more...
Corresponds to a graduate course in mathematics, taught at Carnegie Mellon University in the spring of 1999. This course aims to show that the creation of scientific knowledge is an international enterprise, and who contributed to it, from where, and when. more...
Researchers have been studying complicated classes of problems that can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimization problems. This monograph contains a presentation of the duality theory for these classes of problems and their generalizations. more...
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Contemporary's Number Power: Real World Approach to Math (The Number Power Series)
Book Description: Number Power is the first choice for those who want to develop and improve their math skills. Every Number Power book targets a particular set of math skills with straightforward explanations, easy-to-follow, step-by-step instruction, real-life examples, and extensive reinforcement exercises. Use these texts across the full scope of the basic math curriculum, from whole numbers to pre-algebra and geometry. Number Power: Review builds critical-thinking skills and reviews computational skills from whole numbers to beginning algebra and geometry
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Build algebra understanding with VersaTiles Algebra Readiness
Levels 6-10
Topics include—
Integers, Equations & Inequalities
Rational Numbers, Proportions, & Similarity
Linear & Nonlinear Functions
Two- & Three-Dimensional Spaces
Data Analysis & Probability
Lab includes 25 Student Activity Books (five copies each of five titles), a comprehensive Teacher's Guide, five Answer Cases (or 15 Mini Answer Cases), a storage and carrying case, and also available Algebra Readiness eVersaTiles CD-ROM.
Starter Sets include one copy each of five Student Activity Books, a comprehensive Teacher's Guide, one Answer Case (or 2 Mini Answer Cases), and a zippered storage portfolio.
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This book focuses on essential knowledge for teachers about proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas—essential understandingsThe study of geometry—whether taught as a stand-alone or as a series of topics integrated within other courses—develops core ideas, concepts, and habits of mind that students will need as users of mathematics and as lifelong learners.
The teaching and learning of mathematics involves far more than memorizing procedures and applying algorithms. Problem solving, conjecturing, constructing and critiquing arguments, and communicating and representing mathematical ideas are at the heart of what we are trying to achieve in the classroom. The 2012 Focus Issue of MTMS provides examples and ideas for teachers to implement in their classroom toward these goals.
A valuable resource to any mathematics teacher, this rich collection of mathematical tasks will enliven students' engagement in mathematical thinking and reasoning and help them succeed in the classroom.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Product Description
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn about the history of problem solving and the derivation of the algebraic equation by functional exploration and by symbolic manipulation. Grades 5-9. 30 minutes on DVD. Finally learn the language you've always wanted to learn with the Living Language Method!
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The Math & Finance course is designed to prepare you for both college level business programs and to help you understand our complex financial world. This class will use the basic math skills of addition, subtraction, multiplication and division. Students may either take this class as their third math credit or as a business elective. Due to the nature of the course, students planning on attending college should consider taking a more challenging math course. Most colleges will not recognize this class as a math class.
** If you are a junior taking this course, you should considering taking a fourth year of math!
Supplies Needed:
1. 3-ring binder(1 ½ to 2 inch)
2. 10 subject dividers for binder
3. Calculator
You will need to bring a calculator to class everyday!!!
A 4-function calculator is all that is needed, but it may be any other type as well.
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The Sixth Grade Mathematics course is the first in a series of three integrated Mathematics Courses that meet the NJCCCS in grades six to eight.This course emphasizes numeric reasoning through fluency and facility with numbers, geometry as a means to solve problems and make sense of a variety of phenomena, and measurement as a tool to quantify a variety of problems. Students also cover topics of algebra as a way to communicate the patterns in mathematics, data-analysis as a means to model a variety of real world situations, probability as a way to quantify chance circumstances, and discrete mathematics to develop methods for organizing and interpreting non-continuous data.Problem solving strategies are developed throughout this course.This course will help students appreciate the value of mathematics and continue to develop tools needed for varied educational options.
Course Overview – Grade 7 Mathematics
The Seventh Grade Mathematics course is the second in a series of three integrated Mathematics Courses that meet the NJCCCS in grades six to eight.This course emphasizes geometry, measurement, and proportionality.Problem solving strategies are developed throughout this course.Work with decimals and percents in real life situations are a major thrust of Course 2 as well as more in-depth study of algebra and functions.This series follows a structured sequence that allows for introduction, reinforcement, and extension of topics needed for success in high school algebra and geometry.This course will help students appreciate the value of mathematics and continue to develop tools needed for varied education options.
Course Overview – Grade 8
The 8th grade course is the third in a series of three integrated Mathematic Courses that meet the standards in grades 6 – 8.The course integrates mathematical topics and prepares students for the first year algebra.Other integrated topics include geometry, measurement, statistics, probability, and proportional reasoning.Problem solving activities and applications are integrated into every chapter.Technology plays an important role in Course 3, in particular the use of the graphing calculator and Excel Spreadsheets.The course culminates the three-year strand that will enable our students to enjoy and appreciate the value of mathematics and be able to succeed in the high school.
Course Overview – Grade 7 Pre-Algebra
Seventh Grade Pre-Algebra is a one-year accelerated course designed to meet the needs of the superior elementary mathematics student.This course meets the NJCCCS in grade eight.Students will learn to set up problems using appropriate operations and to employ a variety of problem solving techniques.The student will receive a thorough preparation for work in algebra and future study in mathematics.Curriculum will be enhanced by the use of technology in every unit.
Course Overview – Algebra
The study of Algebra I helps students build their critical thinking and problem solving skills as well as their understanding of a multitude of algebraic concepts.The jobs in the 21st century require employees to have more enhanced problem solving skills and greater mathematical knowledge.The contemporary Algebra I course also incorporates the NCTM standards.Problem solving, applications, reasoning, geometric models, technology, and various exploratory techniques are developed throughout this course.
Middle School
The course offerings and core texts for the middle school math program are as follows:
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"Innovative introductory text . . . clear exposition of unusual and more advanced topics . . . Develops material to substantial level."--American Mathematical Monthly "Refreshingly different . . . an ideal training ground for the mathematical process of investigation, generalization, and conjecture leading to the discovery of proofs and counterexamples."--American Mathematical Monthly " . . . An excellent textbook for an undergraduate course."--Australian Computer Journal A stimulating view of mathematics that appeals to students as well as teachers, this undergraduate-level text is written in an informal style that does not sacrifice depth or challenge. Based on 20 years of teaching by the leading researcher in graph theory, it offers a solid foundation on the subject. This revised and augmented edition features new exercises, simplifications, and other improvements suggested by classroom users and reviewers. Topics include basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms. 1994 ed. Reprint of the revised and augmented edition, Academic Press, Boston, 1994$14.95
Mathematics for Algorithm and Systems Analysis by Edward A. Bender
S. Gill Williamson Discrete mathematics is fundamental to computer science, and this text covers its ideas and mathematical language. Features counting and listing, functions, decision trees and recursion, and basic concepts of graph theory. read more
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6854868 / ISBN-13: 9781576854860
Algebra Success in 20 Minutes a Day
Math jargon is kept to a minimum as Jund offers a hands-on review of the algebra basics taught in high school, with some common workplace ...Show synopsisMath jargon is kept to a minimum as Jund offers a hands-on review of the algebra basics taught in high school, with some common workplace trigonometry thrown in. A pre-test assessment enables readers to address their weaknesses, and step-by-step demonstrations help them solve specific algebra problems.Hide synopsis
For new algebra students or those seeking a refresher, this book offers a series of simple 20-step lesson plans that emphasize quick learning of practical, essential skills.For new algebra students or those seeking a refresher, this book offers a series of simple 20-step lesson plans that emphasize quick learning of practical, essential skills Algebra Success in 20 Minutes a Day
I have always been terrible at any mathematics, especially algebra. I need to learn more advanced math for college now though so I needed to start from the beginning. This book is so good that I was able to use it by myself and learn every concept presented in it in about a month or two
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I'm getting really bored in my math class. It's radical expression symbol with calculator, but we're covering higher grade material. The concepts are really complicated and that's why I usually doze off in the class. I like the subject and don't want to fail, but I have a real problem understanding it. Can someone guide me?
Due to health reasons you might have not been able to attend a few lectures at school, but what if I can you can have your own little classroom, in the place where you live? In fact, right on the laptop that you are working on? Each one of us has missed some classes at some point or the other during our life, but thanks to Algebrator I've never been left behind. Just like an instructor would explain in the class, Algebrator solves our queries and gives us a detailed description of how it was solved. I used it basically to get some help on radical expression symbol with calculator and fractional exponents. But it works well for all the topics.
Hi, I am in Basic Math and I bought Algebrator a few weeks ago. It has been so much easier since then to do my math homework! My grades also got much better. In short, Algebrator is great and this is exactly what you were looking for!
Algebrator is a remarkable product and is surely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can say that it has made learning math more fun.
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determine if integrating a unit on functions would benefit students. Previous studies have shown that integrating science and mathematics increases students' understanding of certain topics in science. TypicallyEach year thousands of students are tracked into mathematics classes. In these particular classes, students may struggle or find their mathematics skills less academically able than their classmates and give up on the tasks that are introduced to
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MathXL
MathXL ( is an online program made by the textbook publisher. I think that you will find it to be a great learning tool - it is filled with resources from video to step-by-step problem-solving guidance. It also allows you to review your quizzes and homework assignments immediately after you take them so that you can see how you did. While it is a wonderful resource for this course, it does have some quirks, which I will discuss below.
First, though, the computer requirements: you will need access to a computer with the latest version of the Adobe Flash Player (Note that MathXL is installed on the computers at the Computer Commons at each Pima campus, so you can always go there to complete the MathXL portions of the course.)
You will also need an access code to get into MathXL. If you bought the correct textbook package, you already have an access code for MathXL. If, however, you bought the textbook, but not the full package listed in the syllabus, or you purchased a used book, you will need to buy an access code from the MathXL website.
Once you have an access code and have logged in, you will need to enroll in the course. Look for the enroll tab on the 'My Courses' page. This will bring up a drop-down menu from which you can select our campus, which is listed as Pima CC - Comm Campus. Then select your instructor (Joseph Erker) and course (Math 151, College Algebra, Spring 2009). Be sure you select the right course!
After you've logged in and installed the necessary software, you should look for the link "Homework and Tests" on the upper left. From there the current quiz & homework assignment should be available. There are also practice tests for extra exercise and review. Underneath the Homework and Tests link is a link called "Study Plan". In there, you'll find more practice problems on different topics. In this part of MathXL you can get feedback to determine which on topics you need the most review/practice.
Grading: You will notice that MathXL grades your assignments automatically. It is not always 100% accurate! It can also be very picky about the format that it wants an answer submitted in, so be sure to follow instructions carefully. I am happy to review any problem that you feel may have been graded incorrectly - simply send me an email with the assignment name and the problem number, along with the reason you feel that the problem may have been graded incorrectly, and I'll get back to you with an explanation and a decision on the grade. One final note on this subject: MathXL typically specifies the form that an answer should be given in (fraction form, decimal form with the answer rounded to three decimal places, simplified as much as possible, etc.) You need to follow those instructions to the letter, otherwise your answer will be marked as incorrect by MathXL and by me.
The only mandatory parts of the course that you do in MathXL are the quizzes, and the homework assignments. All else in MathXL is there as a resource/study guide. However, I encourage you to take advantage of what is available there, and I hope that you find it helpful.
if you are ready, enter MathXL. That web site is where you get an access code if you don't already have one.
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l1
Course: MATH 1501, Fall 2008 School: Georgia Tech Rating:
Word Count: 3463
Document Preview of reckoning or problem solving. The subject of this course was once called the calculus of innitesimals, and was one calculus among many such as the calculus of probabilities. It has proven to be so useful that nowadays it is simply called the calculus, although one does occasionally run into references to other calculi. Calculus is part of a branch of mathematics known as analysis. And the things one analyses in analysis are functions, and the solutions to equations. Q2: What are functions? A function is simply a rule that assigns a member of an output set, usually called the range, to each member of a given set of possible inputs, usually called the domain. In this class, the domain and range will both generally be subsets of the real numbers, but later in the course we will consider the complex numbers as well. There is no restriction; the input and output sets can be any sets at all. What is important though is that exactly one output is assigned to each input. Any relation between inputs and outputs that doesnt satisfy this requirement is just not a function. Q3: O.K., a function is a rule assigning outputs to inputs but how do we specify the rule? There are many ways to specify functions. If the set of possible inputs the domain is a small enough nite set, one could just list the outputs associated to a given input. For example, here is a function f with domain {0, 1, 2, 3} and range {0, 2, 4, 6} given explicitly by f (0) = 0 f (1) = 2 f (2) = 4 f (3) = 6 The input is shown inside the parentheses, and the output is specied to the right of the equal sign. Q4: What if my domain is innite? If there are innitely many inputs, I would need an innitely long list. If there are innitely many inputs, one denitely cant use a list. Instead, one has to specify what it is that the function does to its inputs. 1
You can see what the function that we specied by a list just above is doing it just doubles the input. That makes sense for any real number as input, and by specifying this action of the function we can avoid an innite list! To do this, we turn to variables and operations. A real variable is simply a named container for a real number. Think of it as a box, with a label on front, and a real number inside. We specify variables by name the label on the box. This picture represents a variable x, with the current contained value being = 3.14169...
= 3.14159...
X
It is standard to use letters like x, y and z for the names of variables. There is actually an interesting story behind the use of x as the standard name for a variable. An operation is something you can do the value; i.e, the real number, inside the box. For instance, you can double it, or you can square it. When we write 2x, this means take the value in the box labeled x, and double it. When we write x2 , this means take the value in the box labeled x, and square it. And so on. This way of thinking about variables and operations is familiar to you if you have done any computer programming, especially in a language like C. In this setting, a variable has a name and a type. The name is associated with an address in memory, and the type tells how big a block of memory, starting from the specied address, is used to hold the value. When you call the variable latter in the program, you get the value stored in that block of memory. We can now specify a function f by giving its domain say, all of the real numbers and the sequence of operations it performs on a given variable x, which could be holding any of the values in the domain. For example, f (x) = 2x or f (x) = x2 or f (x) = 2(1 + x2 )
The last example was the only one involving more than a single operation. Here, there were three: rst we squared the value in x, then added 1 to it, and then doubled that. 2
Q4: Cant I just think of the action of f in the last example as one bigger operation? Yes, you can. But actually it wil turn out to be very useful in general to think of functions as an assembly line: We can think of this function f as an assembly line: A box x comes in at the left containing some value, it is opened and sent to three successive stations, where operations are applied to it. Let the three stations be given by the elementary functions p(x), q(x) and r(x) where p(x) = x2 q(x) = 2 x r(x) = 1 + x
x
f(x)
x
p(x)
q(p(x))
r(q(p(x)))
A basic strategy in the analysis of functions in all of science for that matter is to take complicated functions apart into simpler pieces, and then to analyse them in terms of these simpler constituents. We will do this again and again here. The divideand-conquer principle is fundamental to mathematics and much else besides warfare. The assembly line analogy leads us straight into another important notion the composition of functions. If f and g are two functions, and the domain of g contains the range of f , then we get a new function g f , called g composed with f by dening g f (x) = g(f (x)) . This means take the value in x, put it through f , and then take what comes out, and put it through g. 3
This only makes sense if the output values of f are possible input values for g, so the requirement that the domain of g contains the range of f is crucial. However, in many cases the domain of g will contain all real numbers, and there is no problem. Or, if there is a problem, the domain of f can be restricted so it doesnt produce any output values outside the domain of g. Q5: Is composition the only way to build complicated functions out of simple ones? No, composition is just one way to combine a pair of functions to produce a new function. There are others, more closely related to the familiar arithmetic operations we can perform on numbers. For instance, we can add up the outputs of two functions. The net result can be viewed as a new, more complicated function.
x
f(x)
+
x g(x)
(f + g)(x)
More generally, if f and g are two functions with the same domain, we dene new functions by (f + g)(x) = f (x) + g(x) (f g)(x) = f (x) g(x) f g(x) = f (x)g(x) If moreover 0 is not in the range of g, we dene f f (x) (x) = . g g(x) In this way some very complicated functions can be built out of very simple building blocks. For example, with f (x) = x + 1
2
g(x) = 1/x
h(x) = x2 and
j(x) = sin(x)
1 + sin2 (x2 ) = g f h j f + h j h 1 + sin (x + 1) 4
Q6: O.K., so now I know what functions are, and I understand that being able to take them apart into simple pieces is supposed to help me analyze them, but exactly what sort of analysis will I be doing? Suppose I have a function to analyze. Which specic questions will we be asking about it? There are many kinds of questions, and it is not possible to list them all now. But here are two examples that we can talk about at this time. (1) The maximization question: Given a function f , is there an input x0 so that f (x0 ) f (x) for all other x in the domain of f ? If so, what is f (x0 ), the largest output, and what are all of the input values that produce it? (2) The equation solving question: Given a function f and a value a, are there any input values x so that f (x) = a? That is, are there any solutions of the equation f (x) = a? If so, what are they? The rst of these is an optimization problem, as would be the corresponding question about smallest values. If the function has a nite domain, and is given in the form of an explicit list, as in our rst example, then the problem is solved simply by running down the list. But if the domain is innite, we cannot use a list. We must instead analyse the operations, or assembly line steps, out of which the function is built. Calculus provides methods for this. Likewise with the second problem. If f is given by a list, just look down the list of output values and see if you see a. In the special case a = 0, the solutions are called roots of f . We can always reduce to this case by dening a new function g(x) = f (x) a. Then solutions of f (x) = a are roots of g. Calculus provides powerful methods for nding roots. Q7: What kinds of methods will we use besides taking functions apart into simple pieces? Analysis is part of guring things out, and guring things out is quite literally part of analysis. Drawing gures and using geometrical insight will basic to our strategy of analysis. We can bring in a geometric perspective by turning to the graphs of our functions. The range, or some piece of it, is conventionally drawn on the vertical axis, and the domain, or some piece of it, on the horizontal. For each point in the domain of f , draw the vertical line though that input value on the horizontal axis. Then draw the horizontal line through the corresponding output value on the vertical axis. These two lines meet at a single point, which is a point on the graph of f . The graph of f consists of all the points that obtained are in this way. As a subset of the plane, a graph can be drawn on a sheet of paper, or a computer screen. One way of drawing a graph is to compute the points on it for a large but nite collection of input values, and then to connect the dots. This is tedious to do by hand, but easy on a computer. 5
We can do this with Maple very easily. We will just need a few simple commands. (In fact, a few simple commands will get you very far in this whole course. A tuttorial on these commands, written especially for this course, is available on the web.) The function we will take as our example is: f (x) = x (1 + x2 ) . (1 + x4 )
The following three commands dene this function for further use, graph it on the range 0 x 2, and evaluate it at x = 0.5.
> f:= x*(1+x^2)/(1+x^4); f := > plot(f,x=0..2,y=0..1.5); x ( 1 + x2 ) 1 + x4
1.4
1.2
1
0.8 y
0.6
0.4
0.2
0
0.5
1 x
1.5
2
> subs(x=0.5,f); .5882352941 >
6
Now lets zoom in on the part of the graph near the point (0.5, 0.58823) which is almost on the graph as we found by using the substitution command in Maple. The thing to notice is that now we just see something very simple: a straight line.
0.6
0.55
0.5
0.45
0.4
0.45
0.5 x
0.55
0.6
This is an absolutely central point in this whole course: Up close, the graphs of most reasonable functions look like lines. The escape clause about most reasonable functions is necessary. Try for example f (x) = |x| (which you would type into maple using the absolute value function which is denoted by abs(x)) at the point (0, 0). As you zoom in around this point, the graph doesnt even change; it keeps its kink forever. But at any other point, it does become linear if we zoom in. 7
Q7: Its nice that the functions look like lines up close, but whats so good about lines? Lines are very simple they are described by simple equations. Simplicity is not just good its great!. Any line that is not exactly vertical has a nite slope and y-intercept, and is the graph of a linear function f (x) = mx + b . Here m is the slope, and b is the y-intercept. These two numbers m and b completely specify the linear function f , and the corresponding line that is its graph. Another way to specify a line is to give a point (x0 , y0 ) on the line and the slope m. The function f is given by f (x) = m(x x0 ) + y0 . Notice that f (x0 ) = y0 , as must be the case. Also m(x x0 ) + y0 = mx + (y0 mx0 ) so that the y intercept b is y0 mx0 . Now, consider any point (x0 , y0 ) on the graph of f i.e., any point (x0 , y0 ) with y0 = f (x0 ). Let m be the zoomed in slope i.e., the slope of the line we see when we zoom in on (x0 , y0 ). Then the graph of f (x) = m(x x0 ) + y0 is a line which passes through (x0 , y0 ) and ts the graph of f itself there as much as possible. This line is called the tangent line to the graph of f at (x0 , y0 ). The problem of nding the formula for this line which amounts to the problem of nding the zoomed in slope m since x0 and y0 are known is called, naturally enough, the tangent line problem. It is the one of the most central problems of this course. Q8: Whats so important about this tangent line problem? It is not at all obvious that this should be such a central problem. The greeks missed this point completely. So lets recapitulate. Weve seen that up close most functions at most points, anyhow look like linear functions, and linear functions are very simple. This simplicity in the small is the guiding light in the innitessimal calculus. To see how one could use this simplicity in the small to analyse a problem, lets look at an example. Lets try to answer question (1) for the function f (x) = x 1 + x2 1 + x4
on the domain 0 x 2. The graph appears a bit back in this section of notes. You can see a hump near x = 1 where f takes on the value 1. Zoom in on (1, 1) either using Maple, or the graph a few pages back together with your imaginiation) until you see a line. What is its slope? 8
You presumably found a horizontal line; i.e., zero slope. There is exactly one point on the graph at which the zoomed in slope is zero. And as we will see later, it is only at such points that we have to look for maxima and minima. So nding the zoomed in slope will be the key to solving optimization problems. How about the second problem. Lets try to nd the roots of f (x) = x3 2x 5 . Here is a graph centered on x = 2.
15
10
5
01
1.5
2 x
2.5
3
-5
9
There is exactly one root visible, and it is not too far away from x = 2. However, at x = 2, one easily computes that y = f (x) = 1. Now, the slope of the tangent line to the graph y = f (x) at the point (2, 1) turns out to be 10. (Notice the dierent scales on the x and y axes!) You will soon learn how to compute such slopes, but now we want ot explain what they are good for once you have computed them. Using the above information, one easily works out that the eqaution of the tangent line is y = 10x 21 . Now lets graph the function and the tangent line together on the same graph:
15
10
5
01
1.5
2 x
2.5
3
-5
-10
Now, since the graphs of f and its tangent line are close together near x = 2, the place on the x-axis where the tangent line crosses it must be close to the root of f that we are looking for. The later is hard to nd, since f is cubic, but it is easy to nd the place where the tangent line crosses the x-axis: This is given by solving the linear equation 10x 21 = 0, or x = 21/10 = 2.1. so a pretty good approximate solution of our equation is given by x = 2.1, andConditional Probability, Hypothesis Testing, and the Monty Hall ProblemErnie Croot September 17, 2008On more than one occasion I have heard the comment "Probability does not exist in the real world", and most recently I heard this in the context of
Heart of a Whistle-blower RulesDr. Bo BrinkmanMaterials needed:1. A standard deck of playing cards 2. The deck of whistle-blower cards, provided at http:/ player attempts to score points by help
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Name:GTid:Homework 1 SolutionsProf. Loh CS4290/6290 - Spring 20061. ILP (a) I messed this one up in the originally posted solutions. Check the newsgroup for a more detailed explanation. The main tricky parts that are easy to miss are the inter-
Complex Aperture Objective: Understand the effects of aperture diffraction. There are a variety of different tasks that need to be accomplished. It is your responsibility to integrate all of these tasks into a cohesive report. In all of these calcula
HPI RS4 with stock motor $100. Email me for a couple of pics. Its a RS4 with the stock motor(No radio gear). All of the parts on the car look to be in excellent shape. The dogbones, outdrives, belts and plastic parts look like new. Email fredb@ari
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William Foote Whyte STREET CORNER SOCIETYWhyte's study of urban young men in "Cornerville," an Italian neighborhood in Boston, conducted between 1937 and 1940, is one of the best such studies ever done by a sociologist, especially in the skill with
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Newton Raphson AlgorithmSTA705 Spring 2006Let f (x) be a function (possibly multivariate) and suppose we are interested in determining the maximum of f and, often more importantly, the value of x which maximizes f . The most common statistical app
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NOVEL ADAPTIVE TIME-DOMAIN TECHNIQUES FOR THE MODELING AND DESIGN OF COMPLEX RF AND WIRELESS STRUCTURESA Dissertation Presented to The Academic FacultyBy Nathan BushyagerIn Partial Fulfillment Of the Requirements for the Degree Doctor of Philos
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More About
This Book
just the number crunching — and understand how to perform all pre-calc tasks, from graphing to tackling proofs. You'll also get a new appreciation for how these concepts are used in the real world, and find out that getting a decent grade in pre-calc isn't as impossible as you thought.
Updated with fresh example equations and detailed explanations
Tracks to a typical pre-calculus class
Serves as an excellent supplement to classroom learning
If "the fun and easy way to learn pre-calc" seems like a contradiction, get ready for a wealth of surprises in Pre-Calculus For Dummies!
Related Subjects
Meet the Author
Yang Kuang, PhD, is a professor of mathematics at Arizona State University. He currently serves on the calculus committee where he and other members discuss what and how to teach calculus to students majoring in math and physical sciences. Elleyne Kase is a professional 2012
The best purchase I've ever made this year
I bought this book for Precalc cause I was failing and when I studied with this book I improved a whole lot. This book is not vague, it's really transparent. This book answered questions I had that my textbook in math could not answer.
2 out of 2 people found this review helpful.
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Revise for MEI Structured Mathematics - FP1
Catherine Berry, Sophie Goldie, Richard Lissaman, Charlie Stripp
Summary: Revise for MEI Structured Mathematics has been written by experienced authors and examiners especially for A Level mathematics students and provides the ideal preparation for their exam. The series accompanies the MEI Structured Mathematics textbooks. This Revision Guide covers the full content of the FP1 module. Each topic is put into context in terms of its general application, and its links to other modules in the course. It also contains handy reminders of related topics covered previously, whilst worked solutions guide students through all the necessary steps in solving typical questions.
Features to help students to improve their grades and to ensure exam success include:
'Key Facts' - a summary of the essential points to remember 'Caution' - a guide to the most common exam pitfalls students can avoid.
This Revision Guide also has an accompanying website featuring:
'Test Yourself' - interactive multiple choice questions on every topic, with diagnostic answers which identify any weaknesses or common errors Exam-style questions - covering every topic, these audio-visual 'Personal Tutor' worked examples explain exactly how each type of question should be tackled.
This Revision Guide and website can be used throughout the course, but are also perfect for students to use at home on study leave. These ground-breaking Revision Guides will ensure that students go in to their exam with confidence.
written by experienced authors and examiners especially for MEI A Level mathematics students
ideal preparation for the MEI A Level exam
covers the full content of the Further Pure 1 module
worked solutions show students how to solve typical questions
'Key Facts' summarise the essential points to remember
'Caution' indicates the most common exam pitfalls students can avoid
accompanying website contains audio-visual 'Personal Tutor' worked examples explaining exactly how each type of question should be tackled
can be used throughout the course, but perfect for students to use at home on study leave
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The twofold purpose of this study was to trace prospective and practicing mathematics teachers' understandings of content area reading instruction in relation to domain knowledge in mathematics, and to examine the extent to which online pedagogical mentoring supported the integration of such instruction and knowledge. The design called for two pairs of prospective and practicing mathematics teachers to develop, implement, and reflectively evaluate four lessons. A multilevel mentoring approach was used to leverage the valuing of domain knowledge in mathematics. Course materials, lesson plans, teacher reflections, mentors' feedback, interviews, and case reports were analysed using Bourdieu's concepts of cultural capital, field, and misrecognition. Results indicate that despite the study's focus on prioritising domain knowledge through pedagogical mentoring, the instances in which such knowledge were integrated effectively with reading instruction varied in relation to a mentor's expertise in mathematics. If reading teacher educators are to support prospective and practicing mathematics teachers in content area reading instruction, additional sources of mathematics cultural capital are needed.Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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Abstract:
Are sequences functions? What can't the popular "vertical line test" be applied in some cases to determine if a relation is a function? How does the idea of rate of change connect with simpler ideas about proportionality as well as more advanced topics in calculus? Helping high school students develop a robust understanding of functions requires that teachers understand mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about functions. It is organized around five big ideas, supported by multiple smaller, interconnected ideas--essential understandings. Taking teachers beyond a simple introduction to functions, this book will broaden and deepen their mathematical understanding of one of the most challenging topics for students--and themselves. It will help teachers engage their students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. They will also learn to develop appropriate tasks, techniques, and tools for assessing students' understanding of the topic. This book contains three chapters: (1) Functions: The Big Ideas and Essential Understandings; (2) Connections: Looking Back and Ahead in Learning; and (3) Challenges: Learning, Teaching, and Assessing. A foreword, a preface, an introduction, and a list of references are also included. Note:The following two links
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"Focus in Grade 1: Teaching with Curriculum Focal Points" describes and illustrates learning paths for the mathematical concepts and skills of each grade 1 Focal Point as presented in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. It includes representational supports for teaching and learning that can facilitate understanding, stimulate productive discussions about mathematical thinking, and provide a foundation for fluency with the core ideas. This book also discusses common student errors and misconceptions, reasons the errors may arise, and teaching methods or visual representations to address the errors. Because learning paths cut across grades, some discussion of related Focal Points in Kindergarten and grade 2 have been included to describe and clarify prerequisite knowledge and show how the grade 1 understandings build on what went before. "Focus in Grade 1", one in a series of grade-level publications, is designed to support teachers, supervisors, and coordinators as they develop and refine the mathematics curriculum. Contents include: (1) Introduction; (2) Number and Operations; (3) Geometry, Spatial Reasoning, and Measurement; and (4) Mathematizing: Solving Problems, Reasoning, and Communicating, Connecting, and Representing Ideas in First Grade. Preface, Acknowledgments and References are also included.Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure. This guide provides eight specific recommendations intended to help teachers, principals, and school administrators use Response to Intervention (RtI) to identify students who need assistance in mathematics and to address the needs of these students through focused interventions. The guide provides suggestions on how to carry out each recommendation and explains how educators can overcome potential roadblocks to implementing the recommendations. Each recommendation is rated strong, moderate, or low based on the strength of the research evidence for the respective recommendation. Specific recommendations include: (1) Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk; (2) Committee-selected instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8; (3) Instruction during intervention should be explicit and systematic, and should include models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review; (4) Interventions should include instruction on solving word problems that is based on common underlying structures; (5) Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas; (6) Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts; (7) Monitor the progress of students receiving supplemental instruction and other students who are at risk; and (8) Include motivational strategies in tier 2 and tier 3 interventions. Four appendixes are included: (1) Postscript from the Institute of Education Sciences; (2) About the authors; (3) Disclosure of potential conflicts of interest; and (4) Technical information on the studies. A glossary is included. (Contains 314 footnotes, 12 examples and 7 tables.)Note:The following two links
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Out of the 38 nations studied in the 1999 "Trends in International Mathematics and Science Study" (TIMSS), children in Singapore scored highest in mathematics (National Center for Education Statistics, NCES, 2003). Why do Singapore's children do so well in mathematics? The reasons are undoubtedly complex and involve social aspects. However, the mathematics texts used in Singapore present some interesting, accessible problem- solving methods, which help children solve problems in ways that are sensible and intuitive. Could the texts used in Singapore be a significant factor in children's mathematics achievement? There are some reasons to believe so. In this article, I give reasons for studying the way mathematics is presented in the elementary mathematics texts used in Singapore; show some of the mathematics problems presented in these texts and the simple diagrams that accompany these problems as sense-making aids; and present data from TIMSS indicating that children in Singapore are proficient problem solvers who far outperform U.S. children in problem-solving. (Contains 7 figures.)Note:The following two links
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Describes the University of Georgia's Deans' Forum, a group of approximately 30 faculty members who have engaged in collaborative work on issues of teacher education for over 4 years. Discusses its design, activities, and impact, and elements necessary to sustain it. (EV)
Abstract:
This paper contains five essays that describe various aspects of a collaboration called the Deans' Forum at the University of Georgia. A group of 30 faculty committed to exploring issues such as the nature and quality of instruction in university courses, course and curriculum design, learning theories relevant to college age learners, the role of the university in teacher preparation and enhancement, and the role of the university in the P-16 agenda. The five essays are: "How We Got To Expanding the 'Great Conversation' to Include A&S Faculty" (Jenny Penney Oliver); "Reflections on Our Role in Teacher Education by Two Faculty in the Franklin College of Arts and Sciences at the University of Georgia" (Victoria Davion and Hugh Ruppersburg); "The Deans' Forum: Cross-Career Dialogue in English and English Education" (Sally Hudson-Ross, Christy Desmet, and Stephanie Harrison). "The Collaborative Design of Mathematics Courses for Elementary Education Majors" (Sybilla Beckmann and Denise S. Mewborn); and "Outcomes of the Dean's Forum" (Judith Preissle). (Contains 35 references.) (SM)Note:The following two links
are not-applicable for text-based browsers or screen-reading software.Show
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See What's Inside
Product Description
This book examines the study of geometry in the middle grades as a pivotal point in the mathematical learning of students and emphasizes the geometric thinking that can develop in grades 6–8 as a result of hands-on exploration. An essay on the accompanying CD-ROM describes the van Hiele framework and how it can help improve teaching strategies and assessment. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Never worry about battery running down in a test. ? Easy to read clear function keys combined with the assistance of a step-by-step instruction manual to make learning scientific calculations almost painless!
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To guide you through the Maths iGCSE, our tutor-led revision videos start by developing your basic mathematical skills, and progress you onto more advanced levels of learning as you build your confidence and understanding.
Whether you're studying the Cambridge or Edexcel Maths iGCSE, each tutor-led lesson begins with a clear set of learning objectives so you can easily see how it fits within the overall programme of study.
Each clip is presented by a fully-qualified teacher and, by combining easy to follow content with green-screen technology, you won't even realise you're learning.
Please note. *These videos are streamed on-demand for immediate viewing and require an internet connection. We recommend using a WI-FI connection to avoid charges incurred as a result of exceeding your data allowance.
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Thomas is trying to solve a different type of theorem. He is dedicated to teaching people how to understand math, and he has found that at MCLA he can do just that. His love for math is in teaching others its simplicity.
"Postdoctorate work convinced me that I wasn't cut out for pure research and that I really wanted to teach and have more exposure to students," Thomas said, refering to his work at Texas A&M, where he earned his PhD.
"I was teaching one course a semester and it really wasn't enough," he added.
Thomas earned his undergraduate degree at UMass Amherst, then left the area to earn his master's at Tufts University, then his doctorate at A&M. He left Texas, returned to the area his family calls home and started teaching at MCLA during the Fall 2004 semester.
Thomas teaches calculus, linear algebra, history of math, geometry and several other math classes at the College. He also pursues something that has become a passion for him
"Lately I have been putting a special focus on developing courses for future elementary school teachers," Thomas said. He is currently writing a textbook and planning a curriculum in order to educate the educators.
"There are elementary school teachers out there that really don't know math, or they can kind of do it but they really can't explain why it works," he said.
The Department of Education believes a mastery of mathematics is essential for every student, according to its Web site. Part of the department's goal in achieving what it calls STEM, or science, technology, engineering and mathematics education, is ensuring that future teachers are better prepared to teach the curriculum.
"MCLA lept ahead of the curve because a year and a half ago, the College made it required for all elementary education majors to take a three-course series: I think none of the other colleges have required it," Thomas said.
"I am trying very hard to do things in the class that the students can turn around and do in their own [future] classrooms again," Thomas said.
"Unfortunately today, our primary and secondary schools continue to trail many of our competitors, especially in the key area of math and science," President Obama said during his 2009 visit to Hudson Valley Community College in Troy, New York.
The President often cites mathematics as an essential element needed to build a strong, competitive workforce.
Although Thomas expresses disappointment that many politicians see math only as it is applied to job skills, and that the No Child Left Behind act "teaches for the test," he is enthusiastic about the recent spotlight given to math, and the fact that elementary educators are now held to a standard.
"They made this teacher's test that's coming up now, and you really need to know math to pass it," Thomas said, referring to the Massachusetts Test for Educator Licensure (MTEL).
"I have been working on the three-course series, teaching it for the fourth time and am still tweaking it," Thomas said.
He plans to go on sabbatical next spring to finish his textbook." I want it to be simple but throw in a lot of teaching ideas. There are so many projects I want to take on."
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Class News
-
Course Information
Instructor: Herman Gluck
Lecture:
T, Th 3:00 - 4:30 p.m DRL 4C4
Recitation Sections:
M, W 6:30-8:30 p.m. in DRL 4C6.
During our recitation sessions you will presenting solutions to homework problems (those that you have already worked on the previous week and handed in) in front of the class at the blackboard. The goal is to practice laying out a mathematical argument, communicate it effectively, and be able to spot gaps in proofs.
Homework problems are handed in at lecture and returned to you at recitation after it has been graded. Selected problems from each assignment will be graded in detail while the rest may be graded less in depth.
Presenting Solutions:
The goal in written mathematics is to communicate a formal proof clearly enough that a reader can understand the proof without special ingenuity on their part. One must therefore have an idea of who the hypothetical reader is. For us, the hypothetical reader is someone who hasn't thought about the problem and therefore doesn't know what details will come up in the proof or how they will be addressed. We assume though that the reader knows what we know outside of the particular problem at hand and, likewise, is as competent with the current material as we are. This means they know the definitions and they are comfortable taking facts for granted that have been proven earlier in the course.
It is necessary, then, that when we present the solutions to the class that we engage in a little bit of acting - none of us fits the description of the hypothetical reader! The audience should also act the part, asking questions where the solution is ambiguous or unclear. Of course, if you legitimately don't see a step then definitely ask a question too. Likewise for the speaker, feel free to break character and ask a question. (When standing at the blackboard, it is normal to get confused about what you knew when sitting down - don't be ashamed!).
When you present the problem at the blackboard you should be both speaking and writing your solution. Normally it is a good idea to verbally give an overview or an intuitive idea before you do each step. When you write, you should speak aloud what you are writing, either literally or paraphrasing. To reiterate: you should almost always be speaking, either facing the class (while not writing) or while writing (and speaking what you write). As for what you write, it should be complete sentences and self contained, meaning there should not be an essential step which you spoke but did not write. Make sure to use the whole blackboard instead of erasing a single panel over and over.
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Hello people. How do you guys do math systems/substitution so easily? I just never seem to be able to solve a question without going wrong a couple of times times. Please do not tell me to take extra classes. They are costly and I cannot afford them. Any other suggestion would be more than welcome.
You can find numerous links on the internet if you search the keyword math systems/substitution. Most of the content is however designed for the people who already have some knowledge about this subject. If you are a complete novice, you should use Algebra Buster. Is it easy to understand and very helpful too.
Thanks for the detailed information, this seems great. I wanted something exactly
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Course assessment
You will be assessed by a series of modular examinations taken in January and June each year
Materials
Recommended text books include Advancing Maths for AQA - T Graham et al (Heinemann) for each of the modules stated.
For each module the College loans the recommended text books.
Scientific Calculators are required.
Progress
Qualifications:
GCE AS Level Further Mathematics at the end of year 1
GCE A Level (A2) Further Mathematics at the end of year 2
This GCE A Level is a 2 year programme consisting of 6 units. However, you
can study 3 units for just 1 year and gain a GCE AS Level. You must complete
both years and all 6 units for the full GCE A Level. (On completion of A level
Mathematics and A level Further Mathematics, you will have completed
12 different modules).
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Professional Commentary: As a culminating activity to instruction in functions, linear equations, and proportional reasoning, students are introduced to the mathematics of bicycles. Students pair up to investigate bicycle-related relationships, such as wheel diameter and coasting distance or frame tubing size and weight allowances....
Professional Commentary: Students solve two linear programming problems. The first one, Researching Research Papers, involves three variables; the second, The Busing Problem, involves four variables....
Professional Commentary: Students use matrices and technology to solve the Meadows or Malls problem, a linear programming problem with six variables. Students who have not done linear programming problems before are advised to begin with The Busing Problem before attempting Meadows or Malls....
Professional Commentary: Students are asked to minimize cost while maintaining certain criteria in the blending of two distinct types of gasoline. Students use the graph of a system of linear inequalities to solve this linear programming problem geometrically....
Professional Commentary: Students are asked to minimize waste for a local newspaper, which must cut 48-inch rolls of paper into 25- and 21-inch rolls. Students use the graph of a system of linear inequalities to solve this linear programming problem geometrically....
Professional Commentary: Students are asked to maximize profits for an athletic shoe company that produces two kinds of shoes. This is the kind of product-mix problem that occurs whenever a company produces more than one item....
Professional Commentary: Students are asked to minimize the labor costs of hiring different numbers of workers for different shifts at different hourly wages in a pizza shop. Students use the graph of a system of linear inequalities to solve this linear programming problem geometrically....
Professional Commentary: Students see real-life applications of rational functions and the Pythagorean Theorem, as they investigate the essential concepts of how lenses work to magnify vision, and then build simple telescopes to demonstrate their understanding. Discussion questions, suggestions for assessment, additional activities, interdisciplinary connections, and Internet extensions are included in the lesson plan....
Professional Commentary: Students are asked to solve a quadratic inequality. This multiple-choice question is a "hard" test item used in grade 12 in the 2005 National Assessment of Educational Progress (see About NAEP)....
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Whether
teaching calculus at the introductory or AP level, at a high school or
college, there is no better way to explore this rich study of movement
and change than through dynamic animation. Calculus In Motion™ animations are packaged on a CD and perform equally well on
either the Windows or Macintosh platform. An instruction booklet
is included. The animations described below must be opened by
The Geometer's Sketchpad v4 or v5 (no prior versions),
owned and sold by Key Curriculum Press ( on either
Windows or Macintosh platforms.
Although a detailed instruction manual is included on the CD-ROM (PDF
format), most of the animations can be run successfully using only the
on-screen information.
ARC LENGTH
Develop the idea of arc length using any f(x), parametric, or polar
curve & any number of partitions.
AREA BETWEEN 2 CURVES Sweeping horizontally or vertically, the first animation explains the main idea, then 8 specific examples
follow with changeable intervals, and finally, 2 animations (one for vertical sweeps and one for horizontal)
you can enter any desired curves as well as the boundaries of integration.
DEF. OF A DERIVATIVE
DEF.
OF INTEGRATION
INVERSE FUNCTIONS
Drag h to 0 to see PQ become the
tangent line. See the limit process in action. Also, create the
numerical derivative.
Sweep left or right
to accumulate the integral using standard changeable geometric
shapes. Also vary the start and stop points.
Using animated tangent lines,
compare the derivatives of inverse functions. "Morph" the curves
using sliders.
(*also for precalculus)
GRAPHERS Explore slope using animated tangent lines.
See any desired combination of f ', f '', area, and F. "Morph" each graph using
sliders. A 7th animation (not shown below) allows the user to
enter any
desired function and applies all of the same animated features
to it. (*also for precalculus)
RELATED RATES A click of a button advances time to commence the
action to these classic problems.
Other buttons reveal the values
and graphs of the rates.
RIEMANN SUMS Choose rectangles using left endpoints, right
endpoints, or midpoints; or trapezoids to approximate an
integral for any number partitions from 1 to 80!
Functions can be morphed by dragging sliders, or use the first
page to type in any desired function for f(x).
VOLUMES ON A BASE Visualize these shapes one step at a time. Start
by rotating the xy-plane to horizontal. View a few stationary
slices, then a sweeping slice, and finally, an accumulating slice.
Rotate the solid any time for other viewing angles. Choose from an
assortment of bases and cross-sections.
VOLUMES BY REVOLUTION These animations cover both the disk/washer
technique and the cylindrical shell technique.
Develop the process
by first revolving one lone rectangle. Next, revolve several
rectangles in a region and stack or nest the results.
Finally,
revolve any desired region (bounded by 1 or 2 functions of choice)
on an interval of choice, about any horizontal or vertical axis.
SLOPE FIELDS + EULER'S METHOD To introduce what a slope field is, use the graph
of f ' to see its values controlling a gliding dynamic "slope
column". Snapshots of this column are the
slope field. A tangent segment "pilots" the field to draw f.
Once understood, a different animation allows any differential
equation to be entered and generates the slope field.
Manually follow the field to draw f or use Euler's Method (includes
explanation of E.M. and numerical table of data). Easily adjustable.
LIMITS Explore the ε, ∂ definition of limits.
Evaluate the limits (full, left-hand or right-hand) of any
function (including piece-wise defined) as x →a or as x→±∞
MACLAURIN & TAYLOR SERIES Enter any f(x). Overlay a Maclaurin or
Taylor Series polynomial of degree n & use it to approximate
the value of f(x) at any point t. Vertical gray bands show
where the power series is within a chosen tolerance to f(x).
As n increases, the band widens.
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Find a SoutheasternMajor topics studied include: probability, combinatorics, set theory and graph theory. Set theory is the study of sets, both infinite and finite. Some basic operations of set theory include the union and intersection of sets
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Math 125 - Intermediate Algebra with Applications
Course Description
This course is designed for the intermediate algebra student who plans to continue on to MATH 300, 310, 320, 325, STAT 300, 301, or complete an associate degree. It does not fulfill the prerequisite for MATH 315, 330, or higher numbered math courses. Topics include linear functions, models, systems, and graphs, as well as polynomial, exponential, logarithmic, and quadratic functions. The course emphasizes authentic applications and mathematical models using real-world data.
Student Learning Outcomes
Upon completion of this course, the student will be able to:
identify and solve various types of equations and systems of equations.
factor a variety of polynomials.
collect like terms in simplifying polynomial, exponential, and logarithmic functions.
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About The Cartoon Guide to Calculus
Master cartoonist Larry Gonick has already given readers the history of the world in cartoon form. Now, Gonick, a Harvard-trained mathematician, offers a comprehensive and up-to-date illustrated course in first-year calculus that demystifies the world of functions, limits, derivatives, and integrals. Using clear and helpful graphics--and delightful humor to lighten what is frequently a tough subject--he teaches all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education--a valuable supplement for any student, teacher, parent, or professional.
About The Cartoon Guide to Calculus
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Math Placement and Sequencing
Math Placement is one of the most vexing advising issues with entering students. Students who do not want to continue in nath should know that there is no specific "math requirement"; there is, however, the QDS (Quantitative or Deductive Science) requirement, which, it should be noted, can be filled by a number of non-math courses. (see Science Courses for the Non-Major)
Students who want to take a math course to fulfill their QDS requirement but do not want to pursue math or do not need Math 3 or above for other requirements (for instance, pre-health or chemistry requirements) should consider Math 5 (Exploring Mathematics), Math 6 (Intro to Finite Mathematics) or Math 10 (Introductory Statistics)
Many will have placement through pre-matriculation credit or testing and will be placed into one of the following:
Math 3:
Intro to Calculus
Math 1-2:
Covers in two terms what Math 3 covers in one term, and is supported by the Integrated Academic Support Program (IAS) run out of the Academic Skills Center. If a student successfully completes the Math 1-2 sequence they can continue into Math 8 in the spring.
Math 8:
Calculus of Functions of One and Several Variables
Math 11:
Multivariable Calculus, designed specifically for first-year students who place out of Math 3 and 8 and is offered only in the fall. Note that if a student is placed into 11 or 12 but does not want to take math in the fall, s/he will most likely take Math 8, and this will result in losing a pre-matriculation credit. Such a student should talk with Professor Scott Pauls.
Math 12:
An Honors section of Math 11. Note caution above.
The typical sequence is: Math 3, Math 8, Math 13 (14)
Note that:
Math 4:
Applications of Calculus in Medicine and Biology, is designed for students interested in the Life-Sciences or fulfilling pre-health requirements. (Many health profession schools require two calculus courses and this serves as a second course after Math 3.) It cannot serve as a prerequisite for any other courses in math and has MATH 3 (or its equivalent) as a prerequisite.
Math 12:
is the honors section of Math 11.
Math 13:
Calculus of Vector-Valued Functions, is the course that follows MATH 8, and covers much of the same material as MATH 11 but is not interchangeable with Math 11.
Math 14:
an honors section of Math 13
Math 17:
An Introduction to Mathematics Beyond Calculus, is designed for first-year students with credit for 3 and 8 who are particularly motivated and interested in Math. The aim is to introduce a potential Math major to interesting questions in the discipline of Mathematics bef ore the student undergoes the rigors of the major . After taking 17, a student would likely take Math 13/14 or 11/12 if they have not already done so. While it is possible to take Math 17 without credit for Math 3 and 8, it is likely in the student's best interest to take calculus in their first year and then take Math 17 in the second year. A student who wishes to take Math 17 without credit for Math 3 and 8 should consult directly with Sergei Elizalde, who is the instructor of that course in 2012-13.
"Exemption" (EX)
This will occur in a placement record only for students whose placement is being determined by British A-levels. In this case, an "EX" actually indicates a "recommended placement". In this case, a student should consult directly with Professor Scott Pauls about placement if the student wants to take Math.
Professor of Mathematics Scott Pauls (6-1047) serves as the First-Year Advisor, and is available for consultation. A phone call is the best way to get immediate advice.
Typical Introductory Sequencing of Classes The appropriate course for a first-year student is dependent upon his or her math placement (if any). Characteristic sequences are as follows:
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Finding it difficult to gain real motivation to study for HSC General Mathematics? Formulas don't make sense? Or plainly you just can't see the point?
Think about it – General Mathematics, Topic 1: Financial Mathematics. Face it, it doesn't matter who you are – any one and everyone is affected by the world of money – Credit and Borrowing; Annuities and Loans (FM4 & FM5 sub-topics of the HSC Financial Mathematics course).
For example, what about that car you want? Mum and Dad won't get it for you, so you decide to look into loans, but, are unfamiliar just what a loan does and is.
You'll learn all about this through the HSC General Mathematics course. This is the true definer of the General HSC course. It provides students with a firm understanding of the applicability and necessity of Maths in the real world.
The HSC General Mathematics course covers five major topics
What is important is that every student has the best opportunity to establish a firm foundation of Mathematical language, skills, and ideas for each of the five topics covered in General Mathematics.
At Smart Moves our HSC Tutors provide personalised teaching, so that even you to can figure out the point to Algebra. Let our personable HSC Tutors guide you one-on-one through the HSC General Mathematics course – because we all need a bit of motivation.
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Problems
Low-rank approximation.
PCA and Low-rank approximation
Problems
Principal Components Analysis
We will look at PCA, which is a method for analyzing datasets. The mathematics of PCA leads to the ideas of low-rank approximation and positive matrices.
Properties of the SVD
Monday,
Nov 19,
2012
Problems
in the notes
SVD
Friday,
Nov 16,
2012
We prove SVD.
Problems
5.12.1, 5.12.2, and problems in the notes
Schur and Spectral theorem
Thursday,
Nov 15,
2012
We prove Schur triangularization and Spectral theorem.
Problems
7.5.1, 7.5.2, 7.5.3, 7.5.4, 7.5.8, 7.5.10, 7.5.13
Factorizations: review, SVD, spectral, Schur.
Wednesday,
Nov 14,
2012
Today we discuss without proof three factroizations that play an important role in linear algebra: spectral decomposition, Schur triangulaization, and singular value decomposition (SVD). We make a few observations about these factorizations.
Problems
Verify Cayley-Hamilton for triangular matrices
Test 3
Friday,
Nov 09,
2012
Review day
Thursday,
Nov 08,
2012
Diagonalizable matrices
Wednesday,
Nov 07,
2012
Algebraic and Geometric multiplicities.
Monday,
Nov 05,
2012
Problems
7.2.1, 7.2.2, 7.2.3, 7.2.4, 7.2.5, 7.2.9, 7.2.12, 7.2.17, 7.2.21
Eigenvalues, II
Friday,
Nov 02,
2012
Problems
7.1.5, 7.1.8, 7.1.9, 7.1.18
Eigenvalues, I
Thursday,
Nov 01,
2012
Problems
7.1.1, 7.1.3, 7.1.4
Least-squares
Wednesday,
Oct 31,
2012
A discussion of the least-squares method for fitting functions to data. This topic is covered in 4.6 and 5.14.
Problems
4.6.7, 4.6.9
Quiz and problems
Friday,
Oct 26,
2012
Orthogonal projections
Thursday,
Oct 25,
2012
Finishing upThe URV Factorization and projections
Wednesday,
Oct 24,
2012
We prove the URV factorization. We will then move on to 5.13 and discuss orthogonalOrthogonal Complements
Monday,
Oct 22,
2012
We define orthogonal complements and examine some properties.
Problems
5.11.1, 5.11.3, 5.11.4, 5.11.5, 5.11.6, 5.11.8, 5.11.11, 5.11.13
Test 2
Friday,
Oct 19,
2012
Test 2 review
Thursday,
Oct 18,
2012
Projections and idempotents
Wednesday,
Oct 17,
2012
We look at the connection between projections and idempotents. In particular we show that these two classes of linear transformations are the same and that the range and null space of an idempotent are a pair of complementary subspaces.
Complementary Subspaces
Monday,
Oct 15,
2012
Complementary subspaces will play an important role in the development of linear algebra from this point forward. Especially important is the fact that a pair of complemenary subspaces gives rise to a projection
Problems
5.9.1, 5.9.3, 5.9.4, 5.9.5, 5.9.6, 5.9.8
Discrete Fourier Transform
Friday,
Oct 12,
2012
Problems
5.8.1, 5.8.2, 5.8.3, 5.8.5, 5.8.10
Householder reduction
Thursday,
Oct 11,
2012
Problems
5.7.1, 5.7.2, 5.7.3
Elementary reflectors and projectors
Wednesday,
Oct 10,
2012
Problems
5.7.1, 5.7.2, 5.7.3
Orthogonal and unitary matrices
Friday,
Oct 05,
2012
Problems
5.6.1(b)&(c), 5.6.2, 5.6.3, 5.6.5(a)&(b), 5.6.8(a), 5.6.10, 5.6.13
QR factorization
Thursday,
Oct 04,
2012
Problems
5.5.6, 5.5.8, 5.5.11
Parallelogram law, quiz discussion, Gram-Schmidt wrap-up
Wednesday,
Oct 03,
2012
Gram-Schmidt Orthonormalization
Monday,
Oct 01,
2012
Problems
5.5.1, 5.5.2, 5.5.3, 5.5.5
Reminders
Read about QR factorization
Orthogonal vectors
Friday,
Sep 28,
2012
Problems
5.4.1(b)&(c), 5.4.3, 5.4.4, 5.4.6, 5.4.7, 5.4.8, 5.4.9, 5.4.16
Inner product spaces
Thursday,
Sep 27,
2012
Problems
5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.3.5
Matrix norms
Wednesday,
Sep 26 Thursday
Matrix norms
Monday,
Sep 24 Wednesday
Lagrange Multipliers
Friday,
Sep 21,
2012
Vector norms
Thursday,
Sep 20 Friday
Look up Lagrange multipliers
Vector norms
Wednesday,
Sep 19 Thursday
Try checking that the three functions defined at the end of class are norms
Change of basis and similarity
Monday,
Sep 17,
2012
Problems
4.8.1 , 4.8.2 , 4.8.3 , 4.8.6 , 4.8.8
Reminders
Read chapter 5.1 for Wednesday
Exam 1
Friday,
Sep 14,
2012
Review day
Thursday,
Sep 13,
2012
Quiz 2 explanation. Followed by Q&A. 4.8 not on exam 1
Change of basis and Similarity
Wednesday,
Sep 12,
2012
We tried to figure out what happens to the matrix of a linear transformation when we change bases. Then we tried to see what happens to a vector under a change of basis. You now have four homework problems that relate to this topic.
Problems
4.8.1 , 4.8.2 , 4.8.3 , 4.8.6 , 4.8.8
Reminders
Work through all the problems that have been assigned. We have a test and quiz coming up.
Matrix of a linear transformation
Monday,
Sep 10,
2012
Linear Transformations, part 2
Friday,
Sep 07,
2012
Coordinates, the matrix of linear transformation with respect to a basis.
Problems
4.7.11 , 4.7.12 , 4.7.14 , 4.7.17
Reminders
Read 4.8
Linear Transformations, part 1
Thursday,
Sep 06,
2012
Definition and the matrix of a linear transformation
Reminders
Read 4.7
Problem session
Wednesday,
Sep 05,
2012
Work problems from 4.4.
Reminders
Nothing to read, since you read chapter 4.5 already. You did read it, didn't you?
Basis and Dimension
Tuesday,
Sep 04,
2012
Definition of a basis, dimensions of the four fundamental subspaces, computing a basis for each of these.
Problems
4.4.2 , 4.4.3 , 4.4.4 , 4.4.6 , 4.4.7 , 4.4.8 , 4.4.17 , 4.4.18
Reminders
Read chapter 4.5 for Wednesday
Linear independence
Friday,
Aug 31,
2012
Computing spanning sets for the range and kernel using row reduction. Definition of linear independence and basis.
Review of linear systems
Problems
Reminders
Read chapter 3.7 for Wednesday.
Course Overview
Instructor
Mrinal Raghupathi
About this course
This class is a continuation of SM261. A central theme in linear algebra and matrix analysis is the notion of a matrix factorization. These factorizations have important applications in a wide variety of applications. In this class we look at the QR decompositions, SVD and the spectral theorem. We will develop the linear algebraic machinery needed to appreciate these results
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Power up your resume! You've worked hard for your computing degree; now it's time to take that education and put it to work. Get an edge on the other job applicants with Resumes for Computer Careers. This helpful resource is packed with expert advice on creating concise, stylish resumes that will instantly get you noticed. With this go-to guide you'll:
Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis. Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive. Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.
Mathematical Logic for Computer Science is a mathematics textbook with theorems and proofs, but the choice of topics has been guided by the needs of students of computer science. The method of semantic tableaux provides an elegant way to teach logic that is both theoretically sound and easy to understand. The uniform use of tableaux-based techniques facilitates learning advanced logical systems based on what the student has learned from elementary systems. The logical systems presented are: propositional logic, first-order logic, resolution and its application to logic programming, Hoare logic for the verification of sequential programs, and linear temporal logic for the verification of concurrent programs. The third edition has been entirely rewritten and includes new chapters on central topics of modern computer science: SAT solvers and model checking. There are 150 exercises with answers available to qualified instructors. Documented, open-source, Prolog source code for the algorithms is available at Mordechai (Moti) Ben-Ari is with the Department of Science Teaching at the Weizmann Institute of Science. He is a Distinguished Educator of the ACM and has received the ACM/SIGCSE Award for Outstanding Contributions to Computer Science Education. His other textbooks published by Springer are: Ada for Software Engineers (Second Edition) and Principles of the Spin Model Checker.
Discrete Mathematics for Computer Scientists provides computer science students the foundation they need in discrete mathematics. It gives thorough coverage to topics that have great importance to computer scientists and provides a motivating computer science example for each math topic, helping answer the age-old question, "Why do we have to learn this?" Suitable for either lecture-only or fully-interactive, collaborative course environments Intended for students who have completed, or are simultaneously studying, data structures (CS2) Written by leading academics in the field of computer science.
This is the only official, EC-Council-endorsed CHFI (Computer Hacking Forensics Investigator) study guide. It was written for security professionals, systems administrators, IT consultants, legal professionals, IT managers, police and law enforcement personnel studying for the CHFI certification, and professionals needing the skills to identify an intruder's footprints and properly gather the necessary evidence to prosecute.
Computer and Web jobs account for many of the fastest-growing, best-paying jobs in the world of work. To achieve these lucrative jobs, candidates need an edge one they can easily obtain with the help ofExpert Resumes for Computer and Web Jobs.
This book features an extensive collection of sample resumes and cover letters for computer, Web, and IT professionals. These samples show readers at all levels from entry-level to executive how to create exceptional tools for navigating the job market. In addition, the authors provide sound resume and cover letter writing advice, including a new chapter on how to create and use an electronic resume.
Computing Curricula 2001 (CC2001), a joint undertaking of the Institute for Electrical and Electronic Engineers/Computer Society (IEEE/CS) and the Association for Computing Machinery (ACM), identifies the essential material for an undergraduate degree in computer scienceLight and Skin Interactions immerses you in one of the most fascinating application areas of computer graphics: appearance simulation. The book first illuminates the fundamental biophysical processes that affect skin appearance, and reviews seminal related works aimed at applications in life and health sciences. It then examines four exemplary modeling approaches as well as definitive algorithms that can be used to generate realistic images depicting skin appearance. Originally devised for a Spring School organised by the GAMES Networking Programme in 2009, these lectures have since been revised and expanded, and range from tutorials concerning fundamental notions and methods to more advanced presentations of current research topics. This volume is a valuable guide to current research on game-based methods in computer science for undergraduate and graduate students. It will also interest researchers working in mathematical logic, computer science and game theory
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Mathematics Department
General description of expectations for math courses:
Students taking an online math class should expect to have work assigned on a daily basis (Monday through Friday). Students receive a pacing guide at the beginning of each term that shows them exactly what to complete each day of the semester. All work must be completed by midnight on the date it is listed on the pacing guide.
While taking a minimester math class with our campus, you should plan on spending a minimum of 15 hours per week. Students taking a semester-long math class should plan on spending about 10 hours per week on the class. If you are taking a summer semester course, please plan on spending 20 hours per week.
Frequency of Math Chats:
All instructors will hold at least one chat a week according to the schedule posted in each course. These chats are designed to enhance student mastery of the AKS and to provide an opportunity for remediation, reinforcement, and extension. Students are required to attend these chats or listen to the chat recording within 48 hours.
***Face to Face Requirements:
The only face-to-face requirement for mathematics is the Final Exam and/or End of Course Test at the end of the semester. Select courses also require students to come take a county Interim exam at the midpoint of the term.
We also do reserve the right to have a student come take an assessment in a proctored setting at any point during the semester if the need arises.
Materials Pickup Information:
The following math classes require a textbook, which must be checked out from our campus on our scheduled Materials Pickup time:
If at all possible, students should have regular access to a calculator while taking our online math classes. Graphing calculators are the ideal choice for most of our courses. Unfortunately, our campus does not have calculators that we can provide to our students, but we do have a graphing calculator you can download for free onto your computer.
Please click here to see a sample online math course syllabus and associated due dates calendar.
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