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books.google.com - Besides... II For Dummies
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User Review - msrg - Overstock.com
Algebra for Dummies, like most of the "for Dummies" series, explains concepts in a clear, concise manner. My son and his friends did think the book was helpful in clarifying algebra. Some topics ... Read full review
Review: Algebra II For Dummies (For Dummies (Math & Science))
User Review - Jeff Sylvester - Goodreads
Great resource. Covers higher level mathematics but far beyond in depth and breadth from what I can remember taking in Gr. 12 algebra and trigonometry. If you can master the contents of the Dummies ...Read full review
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Algebra II For Dummies:Book Information - For Dummies Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra problems. This friendly guide shows you ... WileyCDA/ DummiesTitle/ Algebra-II-For-Dummies.productCd-0471775819.html
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About the author (2006)
Mary Jane Sterling has authored Algebra For Dummies, Trigonometry For Dummies, Algebra Workbook For Dummies, Trigonometry Workbook For Dummies, Algebra I CliffsStudySolver, and Algebra II CliffsStudySolver. She taught junior high and high school math for many years before beginning her current 25-year-and-counting career at Bradley University in Peoria, Illinois. Mary Jane enjoys working with her students both in the classroom and outside the classroom, where they do various community service projects. |
Algebra Cheat SheetsAlgebra Cheat Sheets is a set of black-line masters showing examples and procedures for teaching 40 algebraic concepts. Most cheat sheets come in two versions -- one with an example and the second with space for student notes. Comprehensive instructions are given for teachers and/or parents.
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277.7 KB | 75-20 years teaching experience in schools
-interactive and group models as well as individual pace
-Montessori Jr HS teacher and then Principal
-25 years educational software development
-25 years educational manipulative development |
Main menu
Boletin UAL: Online Mathematical Project
Submitted by Boletin on Sun, 02/19/2012 - 10:54
Optional Abstract:
The project we present here is an online free journal on the spreading of Mathematics ( mainly addressed to secondary school students, although also appropriate for a broader audience. It is a local action of the University of Almería (Spain) to promote Mathematics among secondary school students and to strengthen relationships with their Math teachers. Our leitmotiv is to provide a closer, more objective, real and positive image of mathematics and help to improve its perception and assessment. This project also includes visits to secondary schools, where the prize of the journal's mathematical contest is awarded to the winner and talks on popularization of Maths are delivered. The journal, created in 2007, is published in Spanish 3 times an academic course, and its length is around 20-25 pages. More than 30 persons are involved in its edition. Last year we had around 115,000 visits and more than 45,000 downloads.
In Spain, it is well known that secondary school students' motivation to study Science (especially Mathematics) is not very strong. This is a common problem to almost all Europe. Spanish and Andalusian governments have general policies on education to promote the study of Science. They are necessary and convenient. However, we think that local actions in the students' environment, which supplement these policies, are also needed.
For this reason, five years ago we founded at the University of Almería an online free journal called Boletín de la Titulación de Matemáticas de la UAL ( mainly addressed to those students. Our intention is to stimulate their interest in mathematics, to develop their mathematical skills, to complement their background, and to improve their perception and assessment of mathematics. Many aspects, usually out of the syllabus, are treated to provide a closer, more real and positive image of mathematics. Another aim is to encourage a closer contact between secondary schools and our university. Although these are our main goals, we do not discard a broader audience and often include articles appropriate for a general public.
It is not a customary journal or gazette but a virtual place where secondary school students and teachers, undergraduate students and university professors can write about their experience with Mathematics in their classes, read and write short articles on mathematical spreading, participate in quizzes, see real life applications of Mathematics, etc. We do not know of a similar project in Spain, and our journal is now known all over the country.
Boletín is available in a well-known free format from our web page (pdf file), and elaborated with the free software LATEX, commonly used in the scientific world.
This project is a joint experience which is carried out by people working or studying in different levels of the Spanish educational system. There are professors and students from our university and teachers from secondary schools. The secondary school students are, as said, the main target of this project, and they can participate through the mathematical contest that we pose. There exists a closer contact later in the prize award ceremony at the winner's secondary school. In such ceremonies, apart from awarding the prize, we give a talk on Maths and inform about our university degree in Maths, thus strengthening relationships with the winner's secondary school and their Math teachers. No doubt, the contest is the way that leads us from our virtual site to the live contact with the most important part of our audience. Regarding our undergraduate students, they have their own section in the journal, called "Territorio Estudiante" (Student Area).
The journal is published in Spanish 3 times an academic course (October, January and April). Its length is around 20-25 pages and papers published must be short and suitable for a secondary school level. It is divided into 5 main sections, each of which has several subsections:
Interviews, mathematical activities, and news.
From Secondary School to University, containing teaching experiences, learning Maths in a foreign language, solutions to the university entrance exam Maths problems, etc.
A mathematical contest for secondary school students.
Mathematical spreading, including short articles about History of Mathematics, applications of Mathematics, Women and Mathematics, Culture and Mathematics, etc. Books and web pages reviews about Mathematics popularization appear here as well. Mathematical quotations and puzzles are also included.
Student area, made itself by our undergraduate students and where, among other things, they speak about their experiences with mathematical studies and activities and interview Erasmus students and alumni.
The editorial committee consists of 3 main editors and 29 section editors, composed by professors of the University of Almería and teachers from secondary schools of our province, and 4 undergraduate students in charge of the student area. Although our goal is acting in our local environment, since we use a global tool such as the Internet, we usually receive experiences and articles from people from other regions, which is an indicator that Boletín is known out of our province. For example, the Spanish Royal Mathematical Society (RSME) in its nice site DivulgaMAT ( includes a link with the summaries of the Boletín issues.
To spread our project, apart from our web site and visits to secondary schools, we announce the publication of a new issue in the RSME electronic bulletin, collaborate with the local press, use Facebook (group Almería Matemática with more than a hundred members), give out merchandising (pens with our logo, mathematical T-shirts, usb memories, etc.) and publicize it among our guest professors and at the universities we visit for research purposes.
Several facts allow us to claim that this project has a very positive effect. We are very interested in knowing if our efforts to bring the Mathematics closer to young people and teachers in schools are useful. Among the methods to know this we use:
Appearance in the local newspapers to achieve more social relevance (see the additional documentation).
Number of visits that we pay to secondary schools and feedback from them since this quantity points out the interest that the Boletín arouses.
Recognition by the local academic authorities. In 2011, we received the award for Excellence in Teaching Innovation.
In conclusion, with the invaluable help of our journal we have set up a reliable atmosphere between professors from our university and teachers from secondary schools in our region of influence. This has permitted to develop other activities jointly such as meetings about mathematical education (I Jornada del Profesorado de Almería, or the group Almería Matemática in Facebook. Our Maths students at the university benefited from this and there is a growing interest among secondary school students in studying the degree in Mathematics at the University of Almería. |
KS5 Alevel Maths Standards units: mostly calculus
Maths worksheet activities and lesson plans. core 1 equations and quadratics. Unit C1 Linking the Properties and Forms of Quadratics. Students classify quadratic functions according to the properties of their graphs and their algebraic forms. This is part of the "Mostly Calculus" set of materials fr More…om Standards unit: Improving learning in mathematics. To enable learners to: identify different forms and properties of quadratic functions; connect quadratic functions with their graphs and properties, including intersections with axes, maxima and minima.
This is fantastic. I'm using it an in introduction to quadratic graphs, every ability is catered for with the use of TI-nSpire for the weaker students and pen&paper for the top group. They are all so engaged at making connections. Thank you so much, I was about to create something like this, but nowhere near as good! |
Visual and interactive way to thorough understanding and mastering Trigonometry without getting wearied on the very first chapter! Java- and web-based math course includes theoretical concepts, hands-on examples featuring animated graphics and live formulas, problem-solving lessons, and customizable real time tests with solutions and evaluations. |
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 |
Analyze, Visualize, Simulate: Mathematica for University Research
Michael Morrison
In this Wolfram Mathematica Virtual Conference 2011 course, learn why Mathematica is used for academic research with a look at its programming language, support for parallel computing, and multiple publishing and deployment options.
Wolfram founder Stephen Wolfram shares the background and vision of Mathematica, including the personal story of how it came to be and why it's in the right place to make profoundly powerful new things possible.
Get the inside scoop on the newest technologies Wolfram is using to make working with Mathematica easier. This talk from the Wolfram Virtual Conference Spring 2013 gives an overview of the Wolfram Predictive Interface, units support, and Wolfram|Alpha integration.
Get an introduction to the HPC and grid computing functionality in Mathematica in this talk from the Wolfram Virtual Conference Spring 2013. The presentation covers a few examples, discusses applications within education work groups, and explores possible ways to scale across available clusters or multicore machines.
This course uses a series of examples to show how models and simulation results from Wolfram SystemModeler can be visualized in Mathematica. Examples used in this Wolfram Virtual Conference Spring 2013 talk include heat loss, batteries, satellite controls, solenoid circuits, and more.
The Computable Document Format (CDF) brings documents to life with the power of computation. In this video, Conrad Wolfram shares examples and explains why Wolfram is uniquely positioned to deliver this technology.
This course from the Wolfram SystemModeler Virtual Conference 2012 focuses on analyzing model equations and simulation results with Mathematica. You'll also learn about the link between Mathematica and SystemModeler.
This video from the Wolfram SystemModeler Virtual Conference 2012 covers core concepts of the Modelica language, which is used by Wolfram SystemModeler. You'll learn how key principles of the language are used in modeling dynamic systems.
Wolfram SystemModeler can be used to model safety-critical systems. This Wolfram Virtual Conference Spring 2013 talk takes a closer look at an aircraft flap system, showing how component faults can be modeled and how their effects on system behavior can be simulated.
Through multiple examples, this course from the Wolfram SystemModeler Virtual Conference 2012 teaches you how to use SystemModeler to develop models of Complex Systems using drag and drop. You'll also learn how you can seamlessly take your models into Mathematica for simulation analysis and model design.
Methods of accessing Wolfram|Alpha from Mathematica are discussed in this Wolfram Mathematica Virtual Conference 2011 course. Learn how to turn results from Wolfram|Alpha into formatted or raw data and computable code or graphics. |
book
Mathematics for the International Student: Mathematics SL has been written to
embrace the syllabus for the two-year Mathematics SL Course, which is one of the
courses of study in the IB Diploma Programme. It is not our intention to define
the course. Teachers are encouraged to use other resources. We have developed
this book independently of the International Baccalaureate Organization (IBO) in
consultation with many experienced teachers of IB Mathematics. The text is not
endorsed by the IBO.
The second edition builds on the strength of the first edition. Chapters are
arranged to follow the same order as the chapters in our Mathematics HL (Core)
second edition, making it easier for teachers who have combined classes of SL
and HL students.
Syllabus references are given at the beginning of each chapter. The new edition
reflects the Mathematics SL syllabus more closely, with several sections from
the first edition being consolidated in this second edition for greater teaching
efficiency. Topics such as Pythagoras' theorem, coordinate geometry, and right
angled triangle trigonometry, which appeared in Chapters 7 and 10 in the first
edition, are now in the 'Background Knowledge' at the beginning of the book and
accessible as printable pages on the CD.
Changes have been made in response to the introduction of a calculator-free
examination paper. A large number of questions have been added and categorised
as 'calculator' or 'non calculator'. In particular, the final chapter contains
over 150 examination-style questions.
Comprehensive graphics calculator instructions are given for Casio fx-9860G,
TI-84 Plus and TI-nspire in an introductory chapter (see p. 17) and,
occasionally, where additional help may be needed, more detailed instructions
are available as printable pages on the CD. The extensive use of graphics
calculators and computer packages throughout the book enables students to
realise the importance, application, and appropriate use of technology. No
single aspect of technology has been favoured. It is as important that students
work with a pen and paper as it is that they use their calculator or graphics
calculator, or use a spreadsheet or graphing package on computer.
This package is language rich and technology rich. The combination of textbook
and interactive Student CD will foster the mathematical development of students
in a stimulating way. Frequent use of the interactive features on the CD is
certain to nurture a much deeper understanding and appreciation of mathematical
concepts. The CD also offers for every worked example.
is accessed via the CD – click anywhere on any worked example to hear a teacher's voice explain each
step in that worked example. This is ideal for catch-up and revision, or for
motivated students who want to do some independent study outside school hours.
For students who may not have a good understanding of the necessary background
knowledge for this course, we have provided printable pages of information,
examples, exercises, and answers on the Student CD – see 'Background knowledge'
(p. 12). To access these pages, click on the 'Background knowledge' icon when
running the CD.
The interactive features of the CD allow immediate access to our own specially
designed geometry software, graphing software and more. Teachers are provided
with a quick and easy way to demonstrate concepts, and students can discover for
themselves and re-visit when necessary.
It is not our intention that each chapter be worked through in full. Time
constraints may not allow for this. Teachers must select exercises carefully,
according to the abilities and prior knowledge of their students, to make the
most efficient use of time and give as thorough coverage of work as possible.
Investigations throughout the book will add to the discovery aspect of the
course and enhance student understanding and learning. Many investigations are
suitable for portfolio assignments.
In this changing world of mathematics education, we believe that the contextual
approach shown in this book, with the associated use of technology, will
enhance the students' understanding, knowledge and appreciation of mathematics,
and its universal application.
Using the interactive student CD
The interactive CD is ideal for independent study.
Students can revisit concepts taught in class and undertake their own revision
and practice. The CD also has the text of the book, allowing students to leave the
textbook at school and keep the CD at home.
By clicking on the relevant icon, a range of interactive features can be accessed:
For a complete list of all the active links on the Mathematics SL second edition CD,
click here.
Graphics calculator instructions: where additional help may be needed,
detailed instructions are availabkle on the CD, as printable pages. Click on the
relevant icon for TI-nspire, TI-84 Plus or Casio fx-9860G.
SELF TUTOR is an exciting feature of this book.
The icon on
each worked example denotes an active link on the CD.
Simply 'click' on the (or anywhere in the example box) to access the worked example, with
a teacher's voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using
movement and colour on the screen. |
Eola, IL Algebra 2 focus on the understanding of important concepts and identification of common ideas and techniques rather than on methods for solving each individual problem type. Identifying the common threads and connections to what the student already knows helps enormously in student understan |
Algebra - Wikipedia, the free encyclopedia Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. For historical reasons, the word "algebra" has several related ... |
Your web browser may be out-of-date. The current math department website does not work correctly with it. Click here to go to the OLD MATH WEBSITE which will work with your browser but which is no longer actively maintained. This NEW SITE is designed with standards-compliant visual browsers in mind. It will work in any Web-capable device. However, if you're using an older visual browser on your computer (like Netscape 4, for instance) and you're seeing this message, you should consider this a friendly reminder to upgrade.
Math 242 covers two distinct topics. The first topic is the integral
calculus, which completes the story of calculus. By the fundamental
theorem of calculus, the integral is in many senses the inverse to the
derivative. The definite integral is particularly useful for
many applications, including valuation of revenue streams.
The second topic will be multivariable optimization, first the
linear case (which is essentially algebra) and then the general
case which requires multivariable calculus.
Lectures and Sections:
Lectures will cover the same basic material as the book, but often
from a different viewpoint. They will be given in a combination of
powerpoint outline, for concepts, and chalkboard, for doing problems.
Being able to do problems is most important, of course. To let you
focus on the problems while you are in lecture,
we will post the projected material in advance. See the
lecture notes page, as linked above.
Class attendance is not mandatory
and in a big lecture faces get lost in the crowd, but if you do regularly
participate in lecture by asking and answering questions I will remember you,
which could be helpful to your final grade.
Sections will mainly cover homework problems. There will
also be quizzes in many weeks (see the exams page,
as linked above). Some sections will focus on
Excel labs, meeting in a computer classroom.
Assignments and exams:
Briefly, there will be homework due almost every week, and five
quizzes. There will be a midterm exam and a final exam. There will
be four Excel-based assignments. |
If you can describe the term,
read on to the next one; if you cannot,
then look it up in the text (the section
number is shown in brackets).
IMPORTANT
IDEAS Can you explain each of these important
ideas in your own words?
Guidelines for Problem Solving[1.1]
Order of Operations [1.2]
Extended Order of Operations [1.3]
Laws of Exponents [1.3]
Inductive vs Deductive Reasoning [1.3]
Euler Circles [1.3]
Next,
make sure you understand the types of
problems in Chapter 1.
Types of Problems
Use Polya's method to solve a problem.
[1.1]
Use Pascal's triangle as an aid to problem
solving. [1.1]
Answer questions by using inductive reasoning.
[1.2]
Simplify an expression using the order
of operations. [1.2, 1.3]
Distinguish inductive from deductive
reasoning. [1.2]
Use Euler circles to determine the validity
of a syllogism [1.2]
Write out exponential numbers without
using exponents. [1.3]
Write a large or small number in scientific
notation. [1.3]
Use a calculator to answer numerical
questions. [1.3]
Estimate answers to numerical questions.
[1.3]
Simplify numerical problems by using
the laws of exponents. [1.3]
Describe the relative sizes of large
and small numbers. [1.3]
Once again, see if you can verbalize
(to yourself) how to do each of the listed
types of problems. Work all of Chapter
1 Review Questions (whether they
are assigned or not).
Work through
all of the problems before looking at
the answers, and then correct each of
the problems. The entire solution is
shown in the answer section at the back
of the text. If you worked the problem
correctly, move on to the next problem,
but if you did not work it correctly
(or you did not know what to do), look
back in the chapter to study the procedure,
or ask your instructor. Finally, go back
over the homework problems you have been
assigned. If you worked a problem correctly,
move on the next problem, but if you
missed it on your homework, then you
should look back in the book or talk
to your instructor about how to work
the problem. If you follow these steps,
you should be successful with your review
of this chapter.
We give all of the answers to the Chapter
Review questions (not just the odd-numbered
questions), so be sure to check your
work with the answers as you prepare
for an examination. |
Curve *A 38-digit precision math emulator for properly fitting high order polynomials and rationals. *A robust fitting capability for nonlinear fitting that effectively copes with outliers and a wide dynamic Y data range.
Graphically Review Curve Fit Results: Once CurveCompact Calculator - CompactCalc - CompactCalc is an enhanced scientific calculator for Windows with an expression editor.CompactCalc is an enhanced scientific calculator for Windows with an expression editor. It embodies generic floating-point routines, hyperbolic and...Desktop Calculator - DesktopCalc - DesktopCalc is an enhanced, easy-to-use and powerful scientific calculator with an expression editor, printing operation, result history list and integrated help.DesktopCalc is an enhanced, easy-to-use and powerful scientific calculator with an... |
Math of Finance is a course approved by the State Department of Education to count as a math credit if it is taught at the rigor of Algebra I or above. The course teaches about aspects of personal income, wages, and tax deductions. Areas of home buying, renting, bill paying, and auto purchases are also covered. Lastly, the course will conclude with personal retirement accounts, planning for the future, and Federal Income Tax filings.
Algebra A&B
Semesters: 1 or 2
Grade level: 9 - 12 Prerequisite: None Description:
The students will explore expressions, equations, functions and rational numbers, learn basic algebraic procedures for solving equation and inequalities, also learn proportional reasoning, graphing techniques, factoring polynomials, and explore rational expressions and equations. Upon completion of this course students will take a state mandatory Instruction Test and that score will be posted on their transcripts.
This course is designed to reinforce and continue using skills learned in Algebra I. Students will explore the language of algebra in verbal, tabular, graphical, and symbolic forms. Upon completion of this course students will take a state mandatory Instruction Test and that score will be posted on their transcripts.
Precalculus explores topics in algebra, trigonometry, and analytic geometry to prepare the students for calculus. Precalculus provides an extensive review of algebra before introducing functions and their graphs. The use of technology is made in many examples and exercises. All exercises, applications, and examples have been selected for their relevance to calculus.
Calculus challenges students to apply math principles to real-life situations and to develop their capacity for problem solving. The skills and lessons in the course will aid the student in the retention of important concepts, reinforce concepts through practice, apply concept in variety of contexts, promote student collection of data, develop logical thinking skills, and improve usage of math theory in proofs.
This course challenges students to develop their capacity for problem solving. The skills and lessons in the course will aid the student in the retention of important concepts, reinforce concepts through practice, apply concepts in a variety of contexts, promote student collection of data, develop logical thinking skills, and improve usage of math theory in proofs. Upon completion of this course students will take a state mandatory Instruction Test and that score will be posted on their transcripts.
The Stigler Public Schools website is under the supervision of Linda Henderson, Web Page Advisor. The Web Page team each year at Stigler High School is responsible for updating and maintaining all information. This team is also under the direct supervision of Linda Henderson, Web Page Advisor. Any questions or comments regarding our Web Site should be directed, through email, to [email protected]. Any technical questions should be directed, through email, to our technical director [email protected]. We do not believe that anything on this web site is copyright protected. If you find something that is copyrighted, please tell us, and it will be removed as quickly as possible. All content has been checked for suitability. However, the Stigler School District is not responsible for the accuracy, nature, or quality of the information on any page not constructed by the Stigler Web Design Class. |
Manhattan Beach ACT MathAs a radar and satellite systems engineer, algebra is key to understanding basic principles of each subject. I |
"I learned about polynomials, and playing Sudoku was fun." Carmen Gutierrez said in addition to solving challenges she enjoyed playing games on the iPad and computers. "I like math because it's unviversal," Carmen, a 12-year-old seventh-grader at ...
So then…change the task required for bitcoin mining from polynomials to protein folding and reward scientific advancement instead. Jean-Claude Morin. The task is unique to each miner because it must include it's own address and the previous block ...
"I plug integers into polynomials and see what integers I get out," she explained. "This is a question that's really easy to ask, but it's very hard to get our hands on the solution. Over the last several hundred years, this has been a question people ...
For example, the guidelines say, even someone who is studying to teach English will be expected to "perform arithmetic operations on polynomials" and "demonstrate knowledge of the physiology of multicellular organisms." Asked why teaching candidates in ...
After three weeks of working through the examples, I recaptured my ninth grade ability to factor polynomials. And so I waited for the day in the Phd program when a professor would ask me to advance to the board and demonstrate this invaluable talent.
That sort of practical problem solving does not come from factoring polynomials but working real world problems with practical application. Back to 1895! Dennis Elam is an assistant professor at Texas A&M San Antonio and a 1966 graduate of Andrews High ...
May 7, 1988 — Given the opportunity to solve multi-variable polynomials and algorithms, most Americans would probably not pass. They have enough problems balancing checkbooks. Coert Olmstead thrives on the complex problems. For his efforts, the math ...
I have to admit I don't factor many polynomials or solve many quadratic equations in my day to day life. But I do use math. I definitely use algebra. I solve for X. Math is something you do use at work - at the grocery store - in deciding whether to ...
Limit to books that you can completely read online
Include partial books (book previews) |
Mathematics and Statistics
Teaching and Learning Standards
The mission of the Department of Mathematics and Statistics at Bowling Green State University is three-fold:
to sustain a curriculum and programs that meet the intellectual and vocational needs of our students,
to foster a sound teaching environment, and
to provide a setting for professional research in mathematics and in statistics.
To accomplish the teaching and learning aspects of its mission, the Department has set seven goals that, ideally, every student will meet at an appropriate level. These are lofty goals or standards that all faculty will help all students meet. We believe that much will be gained by aspiring to meet these standards:
Each student will become proficient at using the language of mathematics and statistics.
Each student will understand what mathematics and statistics are, how they are done, and how they relate to other disciplines.
Each student will be able to objectively and critically evaluate information and assess performance using mathematical ideas.
Each student will develop an appreciation for the beauty, utility, and impact of mathematics and statistics.
Each student will learn mathematical problem solving techniques and become adept at applying them in novel situations.
Each student will learn to use the appropriate technology to successfully attack a wide variety of mathematical tasks.
Each student will understand the basic ideas, techniques, and results of the areas of mathematics and statistics studied.
Further explanation of each goal:
1. Communication Skills
Each student will become proficient at using the language of mathematics and statistics.
There are two distinct roles to communication: (1) sharing information in oral and written form and (2) receiving information by list pages, etc.). In short, every student must be able to read, write, listen and speak effectively about mathematics and statistics.
2. The Nature of Mathematics
Each student will understand what mathematics and statistics are, how they are done, and how they relate to other disciplines.
To delineate what mathematics is is a question that has perplexed philosophers for centuries, but is a question worth thinking hard about. The field has a distinct nature that students should try to understand. Mathematics pages, etc.). In short, every student must be able to read, write, listen and speak effectively about mathematics and statistics.
3. Valuation of Ideas
Each student will be able to objectively and critically evaluate information and assess performance using mathematical ideas.
A great benefit of mathematics and statistics is that they enable us to understand the world around us. Anyone who has studied mathematics and statistics must realize that information is not all of equal value. What one hears and reads must be analyzed critically, evaluated carefully and judged competently, regardless of whether its source was an acquaintance, a newspaper article, a textbook, a professor, or even something you wrote yourself. On some issues -- but not all -- there is a standard of absolute truth and we must learn to recognize when it can occur and when it does occur.
4. Aesthetic Response
Each student will develop an appreciation for the beauty, utility, and impact of mathematics and statistics.
Society has consistently expected the well-educated individual to have a degree of aesthetic understanding and sensitivity. Professional mathematicians are attracted to the field primarily by its beauty and aesthetic appeal. Students need to understand that mathematics and statistics is often done for its own sake. Moreover it is important to understand that mathematics and statistics have had a significant impact on our society and how mathematics and statistics reflect and effect cultural, political, and human issues. Students should be aware of mathematics and statistics as a human endeavor with a rich and complex history that have done much to benefit mankind.
5. Problem Solving
Each student will learn mathematical problem solving techniques and become adept at applying them in novel situations.
The heart of the mathematical sciences is problems. Posing problems, attacking problems, solving problems -- all of these are crucial. Fortunately there are general guidelines that can be taught and learned about problem solving. It takes practice to master them, but once mastered they apply to many situations, including those outside the mathematical sciences. This skill makes individuals educated in the mathematics and statistics incredibly valuable in business, government and teaching -- as well as good citizens.
6. Technology
Each student will learn to use the appropriate technology to successfully attack a wide variety of mathematical tasks.
No task can be accomplished without using the proper tools. In mathematics and statistics there are a great number of tools which the successful practitioner must be able to use effectively. One must learn to use the available tools, be open to the use of new tools, and know when to use which tool. The entering undergraduate major should quickly learn to use the graphic calculator, a word processor, email, and a computer algebra system. The upper-level undergraduate should be proficient at the use of the library (including computer searches), the internet, mathematical manipulatives, the overhead, and an ever expanding variety of software packages and languages (Logo, Geometer's Sketchpad, spreadsheets, statistical packages, etc.). The graduate student should, in addition, be able to create web pages and to use TeX.
7. Content
Each student will understand the basic ideas, techniques, and results of the areas of mathematics and statistics studied. |
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"Number and Operations" forms the basic math knowledge of a students quantitative abilities. It tests a student's abilities in basic arithmetic in areas such as properties of integers, elementary number theory, fractions/decimals, percentages as well as ratios and proportions. Other thinking and arithmetic skills include the ability to compute and use number sequences, understanding unit conversions as well as logical reasoning skills.
The SSAT requires some knowledge of Algebra. Students should know what Algebra is, its uses as well as how to answer questions involving Algebra. This includes simplifying algebraic expressions, solving simple equations as well as working with word problems involving algebra. Substitution or replacing of variables with numbers and evaluating these expressions are also an integral part of Algebra for the purposes of the SSAT. |
Book Description: Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances what they will teach (concepts and content) with how to teach (processes and communication). As a result, students using Mathematics for Elementary School Teachers leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built |
Algebra Unit 2 Assessment Reflection and Goal
" Now that you have completed two assessments, does your final grade (to-date) represent your learning of our Algebra thus far? Why or Why Not?"
87.5% is my final grade for Algebra. I think this grade DOES represent my learning so far because there are a few topics I'm not very clear about, but otherwise most of them are easy and simple for me to understand.
"The class is half-way into unit 3. What is your current goal for your unit 3 assessment? Please describe your preparation for either the blue or green level assessment"
My goal is to achieve at the very least a 90% on the Unit 3 Assessment by completing as much homework as possible, learning all of the content demonstrated in class and reviewing it at home, and demonstrating positive collaboration with fellow classmates in order to further help myself with understand this unit. |
to "A First Course in Electrical and Computer Engineering"This book was written for an experimental freshman course at the University of Colorado. The course is now an elective that the majority of our electrical and computer engineering students take in the second semester of their freshman year, just before their first circuits course. Our department decided to offer this course for several reasons:
we wanted to pique student' interest in engineering by acquainting them
with engineering teachers early in their university careers and by providing
with exposure to the types of problems that electrical and computer engineers
are asked to solve;
we wanted students entering the electrical and computer engineering
programs to be prepared in complex analysis, phasors, and linear
algebra, topics that are of fundamental importance in our discipline;
we wanted students to have an introduction to a software application
tool, such as MATLAB, to complete their preparation for practical
and efficient computing in their subsequent courses and in their professional careers;
we wanted students to make early contact with advanced topics like
vector graphics, filtering, and binary coding so that they would gain a
more rounded picture of modern electrical and computer engineering.
In order to introduce this course, we had to sacrifice a second semester of Pascal programming. We concluded that the sacrifice was worth making because we found that most of our students were prepared for high-level language computing after just one semester of programming.
We believe engineering educators elsewhere are reaching similar conclusions about their own students and curriculums. We hope this book helps
create a much needed dialogue about curriculum revision and that it leads to
the development of similar introductory courses that encourage students to
enter and practice our craft.
Students electing to take this course have completed one semester of calculus, computer programming, chemistry, and humanities. Concurrently with
this course, students take physics and a second semester of calculus, as well as
a second semester in the humanities. By omitting the advanced topics marked
by asterisks, we are able to cover Complex Numbers through Linear Algebra, plus two of the three
remaining chapters. The book is organized so that the instructor can select
any two of the three. If every chapter of this book is covered, including the
advanced topics, then enough material exists for a two-semester course.
The first three chapters of this book provide a fairly complete coverage
of complex numbers, the functions ex and ejθejθ, and phasors. Our department
philosophy is that these topics must be understood if a student is to succeed
in electrical and computer engineering. These three chapters may also be
used as a supplement to a circuits course. A measured pace of presentation,
taking between sixteen and eighteen lectures, is sufficient to cover all but the
advanced sections in Complex Numbers through Phasors.
The chapter on "linear algebra" is prerequisite for all subsequent chapters.
We use eight to ten lectures to cover it. We devote twelve to sixteen lectures
to cover topics from Vector Graphics through Binary Codes. (We assume a semester consisting
of 42 lectures and three exams.) The chapter on vector graphics applies the linear algebra learned
in the previous chapter to the problem of translating, scaling, and rotating
images. "Filtering" introduces the student to basic ideas in averaging and
filtering. The chapter on "Binary Codes" covers the rudiments of binary coding, including Huffman
codes and Hamming codes.
If the users of this book find "Vector Graphics" through "Binary Codes" too confining, we encourage them to supplement the essential material in "Complex Numbers" through "Linear Algebra"
with their own course notes on additional topics. Within electrical and computer engineering there are endless possibilities. Practically any set of topics that can be taught with conviction and enthusiasm will whet the student's
appetite. We encourage you to write to us or to our editor, Tom Robbins,
about your ideas for additional topics. We would like to think that our book
and its subsequent editions will have an open architecture that enables us to
accommodate a wide range of student and faculty interests.
Throughout this book we have used MATLAB programs to illustrate key
ideas. MATLAB is an interactive, matrix-oriented language that is ideally
suited to circuit analysis, linear systems, control theory, communications, linear algebra, and numerical analysis. MATLAB is rapidly becoming a standard
software tool in universities and engineering companies. (For more information about MATLAB, return the attached card in the back of this book to The
MathWorks, Inc.) MATLAB programs are designed to develop the student's
ability to solve meaningful problems, compute, and plot in a high-level applications language. Our students get started in MATLAB by working through
"An Introduction to MATLAB," while seated at an IBM PC
(or look-alike) or an Apple Macintosh. We also have them run through the
demonstration programs in "Complex Numbers". Each week we give three classroom
lectures and conduct a one-hour computer lab session. Students use this lab
session to hone MATLAB skills, to write programs, or to conduct the numerical experiments that are given at the end of each chapter. We require that
these experiments be carried out and then reported in a short lab report that
contains (i) introduction, (ii) analytical computations, (iii) computer code,
(iv) experimental results, and (v) conclusions. The quality of the numerical
results and the computer graphics astonishes students. Solutions to the chapter problems are available from the publisher for instructors who adopt this text for classroom use.
We wish to acknowledge our late colleague Richard Roberts, who encouraged us to publish this book, and Michael Lightner and Ruth Ravenel, who
taught "Linear Algebra" and "Vector Graphics" and offered helpful suggestions on the manuscript.
We thank C. T. Mullis for allowing us to use his notes on binary codes to
guide our writing of "Binary Codes". We thank Cédric Demeure and Peter Massey
for their contributions to the writing of "An Introduction to MATLAB" and "The Edix Editor". We thank Tom Robbins,
our editor at Addison-Wesley, for his encouragement, patience, and many
suggestions. We are especially grateful to Julie Fredlund, who composed this
text through many drafts and improved it in many ways. We thank her for
preparing an excellent manuscript for production.
L. L. Scharf
R. T. Behrens
Boulder, Colorado
To the Teacher:
An incomplete understanding of complex numbers and phasors handicaps
students in circuits and electronics courses, and even more so in advanced
courses such as electromagnetics. optics, linear systems, control, and communication systems. Our faculty has decided to address this problem as early
as possible in the curriculum by designing a course that drills complex numbers and phasors into the minds of beginning engineering students. We have
used power signals, musical tones, Lissajous figures, light scattering, and RLC
circuits to illustrate the usefulness of phasor calculus. "Linear Algebra" through "Binary Codes"
introduce students to a handful of modern ideas in electrical and computer
engineering. The motivation is to whet students' appetites for more advanced
problems. The topics we have chosen – linear algebra, vector graphics, filtering, and binary codes – are only representative.
At first glance, many of the equations in this book look intimidating to
beginning students. For this reason, we proceed at a very measured pace. In
our lectures, we write out in agonizing detail every equation that involves a
sequence or series. For example, the sum
∑n=0N-1zn∑n=0N-1zn
is written out as
1+z+z2+⋯+zN-1,1+z+z2+⋯+zN-1,
(1)
and then it is evaluated for some specific value of z before we derive the
analytical result 1-zN1-z1-zN1-z Similarly, an infinite sequence like limn→∞(1+xn)nlimn→∞(1+xn)n is written out as
and then it is evaluated for some specific x and for several values of n before
the limit is derived. We try to preserve this practice of pedantic excess until
it is clear that every student is comfortable with an idea and the notation for
coding the idea.
To the Student:
These are exciting times for electrical and computer engineering. To celebrate its silver anniversary, the National Academy of Engineering announced
in February of 1990 the top ten engineering feats of the previous twenty-five
years. The Apollo moon landing, a truly Olympian and protean achievement,
ranked number one. However, a number of other achievements in the top
ten were also readily identifiable as the products of electrical and computer
engineers:
communication and remote sensing satellites,
the microprocessor,
computer-aided design and manufacturing (CADCAM),
computerized axial tomography (CAT scan),
lasers, and
fiber optic communication.
As engineering students, you recognize these achievements to be important
milestones for humanity; you take pride in the role that engineers have played
in the technological revolution of the twentieth century.
So how do we harness your enthusiasm for the grand enterprise of engineering? Historically, we have enrolled you in a freshman curriculum of mathematics, science, and humanities. If you succeeded, we enrolled you in
an engineering curriculum. We then taught you the details of your profession
and encouraged your faith that what you were studying is what you must
study to be creative and productive engineers. The longer your faith held,
the more likely you were to complete your studies. This seems like an imperious approach to engineering education, even though mathematics, physics,
and the humanities are the foundation of engineering, and details are what
form the structure of engineering. It seems to us that a better way to stimulate
your enthusiasm and encourage your faith is to introduce you early in your
studies to engineering teachers who will share their insights about some of the
fascinating advanced topics in engineering, while teaching you the mathematical and physical principles of engineering. But you must match the teacher's commitment with your own commitment to study. This means that you must
attend lectures, read texts, and work problems. You must be inquisitive and
skeptical. Ask yourself how an idea is limited in scope and how it might be
extended to apply to a wider range of problems. For, after all, one of the great
themes of engineering is that a few fundamental ideas from mathematics and
science, coupled with a few principles of design, may be applied to a wide
range of engineering problems. Good luck with your |
Thursday: **Exam Review—work first ½ of review. Students will check their answers and ask questions
Reinforce/Review
x
x
x
x
x
Other
Friday: **Exam Review—work second ½ of review. Students check their work and ask questions.
II. LEARNER OBJECTIVES AND PROCEDURES:
The Student Will: (IMP Learner *)
M
TU
W
TH
F
Monday:1a: Simplify radical expressions
Take Notes
x
x
x
x
X
Examine & Discuss
x
x
x
x
x
Tuesday: 1a: Simplify radical expressions
Perform Lab Exercise
Complete Class work
x
x
x
x
X
Wednesday: 1a: Simplify radical expressions
Check Homework Assign.
X
x
x
Read & Discuss
Thursday: Review for exam
Demonstrate
x
x
x
x
X
Take Test
Friday: Review for exam
OralReading
Other
**Lesson plans/assignments will be modified to accommodate IEP provisions.
BROOKHAVEN HIGH SCHOOL
Week Of
May 6x
X
Monday: ****Continue going over SATP test #4. Work each problem—play special attention to those missed by the majority of students. Ask/answer questions. Give calculator tips and other ways to work problem.
Lecture
X
x
Discuss
X
x
x
x
Tuesday: ****Continue going over SATP test #4. Work each problem—play special attention to those missed by the majority of students. Ask/answer questions. Give calculator tips and other ways to work problem.
Demonstrate
x
x
x
X
Question/Answer
X
x
x
x
Wednesday: **ALGEBRA I
STATE TEST DAY!!!!!
Give Examples
x
x
X
x
Read
Thursday: ** Notes 11.6
Page 809 #24-60 even and #61
Reinforce/Review
x
x
x
X
Other
SATP
Friday: ** Notes 11.7
Pages 813-814 #16-46 evenx
x
x
Tuesday: Prepare for SATP
Score Proficient or Advanced!!!
Perform Lab Exercise
Complete Class work
x
x
x
X
Wednesday:SATP ALGEBRA I TEST DAY
Check Homework Assign.
X
Read & Discuss
x
X
x
x
Thursday: 1a: Simplify radical expressions
Demonstrate
x
x
x
X
Take Test
SATP
Friday: 1a: Simplify radical expressions
OralReading
Other
**Lesson plans/assignments will be modified to accommodate IEP provisions.
BROOKHAVEN HIGH SCHOOL
Week Of
April 29Give back and go over Homework #4
Lecture
Discuss
X
x
X
Tuesday: **Homework #4 QUIZ
Demonstrate
X
x
x
Question/Answer
X
x
X
Wednesday: **Students complete Practice test #4 that was started last Friday.
Give Examples
X
x
x
Read
Thursday: **.Discuss first ½ of test #4PT
Friday: **Continue going over SATP test #43A: Apply the concept of slope to determine if lines are parallel or perpendicular. 3B: Solve problems interpreting slope as rate of change.
Take Notes
x
x
X
Examine & Discuss
X
x
x
Tuesday: 3A: Apply the concept of slope to determine if lines are parallel or perpendicular. 3B: Solve problems interpreting slope as rate of change.
Perform Lab Exercise
Complete Class work
X
x
X
Wednesday: Prepare for SATP
Score Proficient or Advanced!!!
Check Homework Assign.
x
Read & Discuss
x
X
Thursday: Prepare for SATP
Score Proficient or Advanced!!!
Demonstrate
X
x
x
Take Test
Q
PT3 in class. Work problems together. Answer/ask questions
Demonstrate
X
X
x
x
Question/Answer
X
x
x
X
Wednesday: **QUIZ on homework #3
Give Examples
X
X
x
x
Read
Thursday: **ContinueX
x
x
Tuesday:2Perform Lab Exercise
Complete Class work
X
x
x
X
Wednesday: 2Check Homework Assign.
X
x
Read & Discuss
x
x
X
Thursday: Prepare for SATP
Score Proficient or Advanced!!!
Demonstrate
X
X
x
x
Take Test
Q2 in class. Work problems together. Answer/ask questions
Demonstrate
X
x
x
Question/Answer
x
x
X
Wednesday: **QUIZ on homework #2
Give Examples
X
x
x
Read
Thursday: **BeginPTX
Examine & Discuss
X
x
x
Tuesday:2a: Solve, check and graph multi-step linear equations and inequalities in one variable.
**Lesson plans/assignments will be modified to accommodate IEP provisions.
BROOKHAVEN HIGH SCHOOL
Week Of
April 1Student Holiday/Professional Development
Lecture
Discuss
x
x
X
X
Tuesday: ** Continue practice test #2. Call attention to the common mistakes made by students. Stress use of highlighter and give calculator tips. Pay particular attention to the problems with parallel & perpendicular lines Give students Homework #1 (due Friday)
Demonstrate
x
x
x
X
Question/Answer
X
x
x
x
Wednesday: ** Continue practice test #2. Students will work #37-55 with a partner. Teacher will assist as necessary. Remind students that homework #1 is due on Friday. Answer any questions about homework #1
**Lesson plans/assignments will be modified to accommodate IEP provisions.
BROOKHAVEN HIGH SCHOOL
Week Of
March 18, 2013
WEEKLY LESSON PLAN
Teacher
V. Brown
R. Coker
K. Showers
S. Wallace
Subject
Algebra I (period) SATP practice test #1
Students will work the practice test individually.
Lecture
Discuss
X
x
x
x
X
Tuesday: continue SATP #1
Students will work the practice test individually
Demonstrate
x
X
x
x
x
Question/Answer
x
x
x
x
X
Wednesday: continue SATP #1
Students will work the practice test individually
Give Examples
X
x
x
x
x
Read
Thursday: continue SATP #1
As soon as students have completed the practice test #1, the teacher will mark any incorrect answers. The grade will be computed for informative purposes only. We will begin going over each problem as a class.
Reinforce/Review
x
x
x
x
x
Other
Friday: continue SATP #1
II. LEARNER OBJECTIVES AND PROCEDURES:
The Student Will: (IMP Learner *)
M
TU
W
TH
F
Monday: Prepare for SATP
Take Notes
x
x
x
x
X
Examine & Discuss
X
x
x
x
x
Tuesday: Prepare for SATP
Perform Lab Exercise
Complete Class work
x
x
x
x
X
Wednesday: Prepare for SATP
Check Homework Assign.
Read & Discuss
Thursday: Prepare for SATP
Demonstrate
x
x
x
x
x
Take Test
Friday: Prepare for SATP
OralReading
Other
BROOKHAVEN HIGH SCHOOL
Week Of
February 25Begin SATP review guide PreTest—students will work on this like it is the real test. No teacher assistance.
Lecture
Discuss
x
x
X
Tuesday: **Continue SATP review guide PreTest
Demonstrate
X
x
x
Question/Answer
x
x
X
Wednesday: **Complete PreTest and/or begin to go over as a class.
(Students will receive a grade for the test which should be curved.)
Give calculator and test-taking tips as class is going over PreTest.
Give Examples
X
x
x
Read
Thursday: **Continue going over the PreTest as a class. Teacher will answer/ask questions based on the individual needs in each period.
Give calculator and test-taking tips as class is going over PreTest
Reinforce/Review
X
x
x
Other
PT
PT
PT
Friday: ** Continue going over the PreTest as a class. Teacher will answer/ask questions based on the individual needs in each period.
Give calculator and test-taking tips as class is going over PreTest and review for Benchmark Exams
Take Notes
x
x
X
Examine & Discuss
X
x
x
Tuesday: Prepare for SATP and review for Benchmark Exams
Perform Lab Exercise
Complete Class work
x
x
x
x
X
Wednesday: Prepare for SATP and review for Benchmark Exams
Check Homework Assign.
Read & Discuss
X
x
x
Thursday: Prepare for SATP and review for Benchmark Exams
Demonstrate
x
x
x
x
X
Take Test
PT
PT
PT
Friday: Prepare for SATP and review for Benchmark Exams
OralReading
Other
**Lesson plans/assignments will be modified to accommodate IEP provisions.
BROOKHAVEN HIGH SCHOOL
Week Of
February 18State test Samples #3—students work by themselves, then we will go over in class.
Lecture
Discuss
x
x
x
X
Tuesday: **STS #4 ½--students will work in groups. Teacher will assist as necessary. Teacher will give correct answers and address questions before students leave.
Demonstrate
X
x
x
x
Question/Answer
x
x
x
X
Wednesday: **STS #6 ½--Students work individually, then go over together. Give students STS #7 ½. Instruct them to work the front for homework.
Give Examples
X
x
x
x
Read
Thursday: **STS # 7 ½ --Students work in groups to complete the back of the page. Teacher will check the front as they work. Teacher will also address any questions. Give correct answers/answer questions before they leave. |
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors).
University Calculus, Early Transcendentals, Second Edition is the ideal choice for professors who want a streamlined text with plenty of exercises. This text helps students successfully generalize and apply the key ideas of calculus through clear and precise explanations, thoughtfully chosen examples, and superior exercise sets. This text offers the right mix of basic, conceptual, and challenging exercises, along with meaningful applications. This significant revision features more examples, more mid-level exercises, more figures, improved conceptual flow, and the best in technology for learning and teaching.
The text is available with a robust MyMathLab® course–an online homework, tutorial, and study solution designed for today's students. In addition to interactive multimedia features like Java™ applets and animations, thousands of MathXL® exercises that reflect the richness of those in the text are available for students.
Table of contents
1. Functions
1.1 Functions and Their Graphs
1.2 Combining Functions; Shifting and Scaling Graphs
1.3 Trigonometric Functions
1.4 Graphing with Calculators and Computers
1.5 Exponential Functions
1.6 Inverse Functions and Logarithms
2. Limits and Continuity
2.1 Rates of Change and Tangents to Curves
2.2 Limit of a Function and Limit Laws
2.3 The Precise Definition of a Limit
2.4 One-Sided Limits
2.5 Continuity
2.6 Limits Involving Infinity; Asymptotes of Graphs
3. Differentiation
3.1 Tangents and the Derivative at a Point
3.2 The Derivative as a Function
3.3 Differentiation Rules
3.4 The Derivative as a Rate of Change
3.5 Derivatives of Trigonometric Functions
3.6 The Chain Rule
3.7 Implicit Differentiation
3.8 Derivatives of Inverse Functions and Logarithms
3.9 Inverse Trigonometric Functions
3.10 Related Rates
3.11 Linearization and Differentials
4. Applications of Derivatives
4.1 Extreme Values of Functions
4.2 The Mean Value Theorem
4.3 Monotonic Functions and the First Derivative Test
4.4 Concavity and Curve Sketching
4.5 Indeterminate Forms and L'Hôpital's Rule
4.6 Applied Optimization
4.7 Newton's Method
4.8 Antiderivatives
5. Integration
5.1 Area and Estimating with Finite Sums
5.2 Sigma Notation and Limits of Finite Sums
5.3 The Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Indefinite Integrals and the Substitution Rule
5.6 Substitution and Area Between Curves
6. Applications of Definite Integrals
6.1 Volumes Using Cross-Sections
6.2 Volumes Using Cylindrical Shells
6.3 Arc Length
6.4 Areas of Surfaces of Revolution
6.5 Work
6.6 Moments and Centers of Mass
7. Integrals and Transcendental Functions
7.1 The Logarithm Defined as an Integral
7.2 Exponential Change and Separable Differential Equations
7.3 Hyperbolic Functions
8. Techniques of Integration
8.1 Integration by Parts
8.2 Trigonometric Integrals
8.3 Trigonometric Substitutions
8.4 Integration of Rational Functions by Partial Fractions
8.5 Integral Tables and Computer Algebra Systems
8.6 Numerical Integration
8.7 Improper Integrals
9. Infinite Sequences and Series
9.1 Sequences
9.2 Infinite Series
9.3 The Integral Test
9.4 Comparison Tests
9.5 The Ratio and Root Tests
9.6 Alternating Series, Absolute and Conditional Convergence
9.7 Power Series
9.8 Taylor and Maclaurin Series
9.9 Convergence of Taylor Series
9.10 The Binomial Series and Applications of Taylor Series
10. Parametric Equations and Polar Coordinates
10.1 Parametrizations of Plane Curves
10.2 Calculus with Parametric Curves
10.3 Polar Coordinates
10.4 Graphing in Polar Coordinates
10.5 Areas and Lengths in Polar Coordinates
10.6 Conics in Polar Coordinates
11. Vectors and the Geometry of Space
11.1 Three-Dimensional Coordinate Systems
11.2 Vectors
11.3 The Dot Product
11.4 The Cross Product
11.5 Lines and Planes in Space
11.6 Cylinders and Quadric Surfaces
12. Vector-Valued Functions and Motion in Space
12.1 Curves in Space and Their Tangents
12.2 Integrals of Vector Functions; Projectile Motion
12.3 Arc Length in Space
12.4 Curvature and Normal Vectors of a Curve
12.5 Tangential and Normal Components of Acceleration
12.6 Velocity and Acceleration in Polar Coordinates
13. Partial Derivatives
13.1 Functions of Several Variables
13.2 Limits and Continuity in Higher Dimensions
13.3 Partial Derivatives
13.4 The Chain Rule
13.5 Directional Derivatives and Gradient Vectors
13.6 Tangent Planes and Differentials
13.7 Extreme Values and Saddle Points
13.8 Lagrange Multipliers
14. Multiple Integrals
14.1 Double and Iterated Integrals over Rectangles
14.2 Double Integrals over General Regions
14.3 Area by Double Integration
14.4 Double Integrals in Polar Form
14.5 Triple Integrals in Rectangular Coordinates
14.6 Moments and Centers of Mass
14.7 Triple Integrals in Cylindrical and Spherical Coordinates
14.8 Substitutions in Multiple Integrals
15. Integration in Vector Fields
15.1 Line Integrals
15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
15.3 Path Independence, Conservative Fields, and Potential Functions
15.4 Green's Theorem in the Plane
15.5 Surfaces and Area
15.6 Surface Integrals
15.7 Stokes' Theorem
15.8 The Divergence Theorem and a Unified Theory
16. First-Order Differential Equations (Online)
16.1 Solutions, Slope Fields, and Euler's Method
16.2 First-Order Linear Equations
16.3 Applications
16.4 Graphical Solutions of Autonomous Equations
16.5 Systems of Equations and Phase Planes
17. Second-Order Differential Equations (Online)
17.1 Second-Order Linear Equations
17.2 Nonhomogeneous Linear Equations
17.3 Applications
17.4 Euler Equations
17.5 Power Series Solutions
Appendices
1. Real Numbers and the Real Line
2. Mathematical Induction
3. Lines, Circles, and Parabolas
4. Conic Sections
5. Proofs of Limit Theorems
6. Commonly Occurring Limits
7. Theory of the Real Numbers
8. Complex Numbers
9. The Distributive Law for Vector Cross Products
10. The Mixed Derivative Theorem and the Increment Theorem
11. Taylor's Formula for Two Variables
New to this edition
760 new, updated, and improved exercises, including many new mid-level exercises, provide more ways for you to address your students' needs
New examples in many sections clarify or deepen the meaning of the topics covered and help students understand their mathematical applications and consequences to science and engineering.
75 new figures have been added, and many others have been revised to enhance conceptual understanding.
MyMathLab now has more than 7,000 assignable exercises, including 670 that address prerequisite skills, giving you the selection you need to create the right homework assignments and assessments.
Content revisions are apparent throughout the text, and many of these changes were driven by suggestions from users:
Limits: to improve the flow of the chapter on limits, the authors have combined the ideas of limits involving infinity and their associations with asymptotes to the graphs of functions, placing them in the final section of the chapter.
Differentiation: to coalesce the derivative concept into a single chapter, the authors have moved the section "Tangents and the Derivative at a Point" from the end of Chapter 2 to the beginning of Chapter 3. In addition, they have reorganized and increased the number of the related rates examples in Section 3.10, and have added new examples and exercises on graphing rational functions in Section 4.4.
Integration coverage has been improved. Integrals, as "limits of Riemann sums," motivated primarily by the problem of finding the areas of general regions with curved boundaries, now form the substance of Chapter 5. After carefully developing the integral concept, the authors turn the focus to its evaluation and connection to antiderivatives captured in the Fundamental Theorem of Calculus. The ensuing applications then define the various geometric ideas of area, volume, lengths of paths, and centroids all as limits of Riemann sums giving definite integrals, which can be evaluated by finding an antiderivative of the integrand.
Parametric equations have been moved to Chapter 10, joining polar coordinates and conic sections, in order to better prepare students for coverage of vectors and multivariable calculus.
Series coverage is more accessible to students with the addition of a number of new figures and exercises as well as the revision of some of the proofs related to convergence of power series.
Vector-valued functions coverage has been streamlined to place more emphasis on the conceptual ideas supporting the later material on partial derivatives, the gradient vector, and line integrals.
Multivariable calculus: the authors have reorganized the opening material on double integrals, and combined the applications of double and triple integrals to masses and moments into a single section covering both two- and three-dimensional cases. This reorganization allows for better flow of the key mathematical concepts, together with their properties and computational aspects.
Vector Fields: important theorems and results are stated more clearly and completely together with enhanced explanations of their hypotheses and mathematical consequences. The area of a surface is now organized into a single section, and surfaces defined implicitly or explicitly are treated as special cases of the more general parametric representation. Surface integrals and their applications then follow as a separate section.
Differential equations: The authors give an introductory treatment of first-order differential equations in Chapter 16, including a new section on systems and phase planes, with applications to the competitive-hunter and predator-prey models. An introduction to second-order differential equations is in Chapter 17. Both of these chapters are available for download on the Thomas' Calculus website (
Features & benefits
The exercises are known for their breadth and quality, with carefully constructed exercise sets that progress from skills problems to applied and theoretical problems. The text contains more than 8,000 exercises in all.
End-of-chapter exercises feature review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises.
Figures are conceived and rendered to support conceptual reasoning and provide insight for students. They are also consistently captioned to aid understanding.
The flexible table of contents divides complex topics into manageable sections, allowing instructors to tailor their course to meet the specific needs of their students. For example, the precise definition of the limit is contained in its own section and may be skipped.
Complete and precise multivariable coverage enhances the connections of multivariable ideas with their single-variable analogues studied earlier in the book.
A robust MyMathLab course contains more than 7,000 assignable exercises, a complete e-book, and built-in tutorials so students can get help whenever they need it.
A complete suite of instructor and student supplements saves class preparation time for instructors and improves students' learning.
Author biography
Joel Hass received his PhD from the University of California–Berkeley. He is currently a professor of mathematics at the University of California–Davis. He has coauthored six widely used calculus texts as well as two calculus study guides. He is currently on the editorial board of Geometriae Dedicata and Media-Enhanced Mathematics. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass's current areas of research include the geometry of proteins, three dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking.
Maurice D. Weir holds a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. He is a Professor Emeritus of the Department of Applied Mathematics at the Naval Postgraduate School in Monterey, California. Weir enjoys teaching Mathematical Modeling and Differential Equations. His current areas of research include modeling and simulation as well as mathematics education. Weir has been awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He has coauthored eight books, including the University Calculus series and the twelfth edition of Thomas' Calculus.
George B. Thomas, Jr. (late) of the Massachusetts Institute of Technology, was a professor of mathematics for thirty-eight years; he served as the executive officer of the department for ten years and as graduate registration officer for five years. Thomas held a spot on the board of governors of the Mathematical Association of America and on the executive committee of the mathematics division of the American Society for Engineering Education. His book, Calculus and Analytic Geometry, was first published in 1951 and has since gone through multiple revisions. The text is now in its twelfth edition and continues to guide students through their calculus courses. He also co-authored monographs on mathematics, including the text Probability and Statistics. |
Course Guide
Course Goals:
This course continues math 370 from the spring semester of 2011.
The focus of the course will be on constructing precise and complete mathematical
proofs. Topics will include:
Ring theory
Factorization
Modules
Canonical forms of matrices
Field theory
Galois theory
Some further group theory, as time permits
Texts:
``Algebra" by Michael Artin, 3rd edition. The 2nd edition is also acceptable.
This is the book used in math 370 in the spring of 2011. I would strongly recommend that you get
an account on library.nu. This is an
amazing site for downloading texts of all kinds, including many mathematical
texts which are by now almost impossible to obtain in print. Here is
some additional advice about how to use the library.nu web site:
We'll use Skype during
online office hours as well as during course podcasts. It's possible to carry
on a conference call with 24 people on Skype, and we have only 8 class members
at the moment. So we can in fact have a conference call which includes everyone in class.
Please send an e-mail to me at [email protected] with your Skype name so
that I can put you on conference calls. If you think there may be times when it would be more
convenient to reach you by phone, please also send a phone number I can use.
From time to time, I will put both streaming and downloading video on
the web.
During conference calls, I will be using an online white board
available from skrbl.com. Have a look at this sample
white board. Feel free to write a message on the white board to test it!
(PG-13 messages only, please.) At the start of a conference call I will
send out the address of the whiteboard we will be using that day.
Everyone should be able to write and to draw pictures on the whiteboard.
Office hours:
By appointment and by Skype in the evenings.
How to make attending lectures efficient:
Before each lecture, check the
current homework and lecture schedule,
and read the appropriate parts of the text. After each lecture, you should review your lecture notes,
reread the corresponding sections of the book and solve related
homework problems.
Homework:
Homework will be due at the beginning
of the first class in a given week. There will be two kinds
of problems on the homework.
The first kind, labelled
Type A, are problems you should work on and
write up on your own.
The purpose of these problems is to provide practice
in writing complete proofs and mathematical arguments.
The second kind of problem, labelled Type B, are
problems you can work on with one or two other students.
I will ask people to be prepared to explain at the board
during class the solutions to Type B problems which
they worked out with their group. You will not need
to turn in written solutions to Type B problems. The
purpose of these problems is to get some practice
at explaining mathematical ideas and arguments
to other people.
There will be no class on Monday, May 30, because of the Memorial Day holiday.
So the first homework set will be due Tuesday, May 31, and you
should be prepared to discuss the Type B problems in class on the homework
that day.
Exams:
During the second half of class on Monday, June 13, there will be a one hour mid-term exam.
The final exam will be held during class on Thursday, June 30.
Getting help:
You are very welcome to arrange a time to meet with me either in math department
or online. |
Analysis and Applications publishes high quality mathematical papers that treat those parts of analysis which have direct or potential applications to the physical and biological sciences and engineering. |
Personal tools
Sections
MATLAB
MATLAB is a high-level language and interactive environment for numeric computation.
MATLAB is high-level language and interactive environment for numeric computation created by Cleve Moler in the 1970s. It was initially designed to give students access to the LINPACK and EISPACK linear algebra libraries without having to learn to code in Fortran. MATLAB has now evolved into one of the most widely used numerical computation tools for research and data visualization in math, engineering and science.
MATLAB contains a sophisticated suite of tools for solving differential equations. Though MATLAB has a learning curve for students to overcome, since it is so widely used it can be beneficial to students to learn how to use MALAB.
There are many books on how to use MATLAB to solve differential equations, but perhaps the most authoritative is Solving ODEs with MATLAB, by Shampline, Gladwell and Thompson. |
Calc II Suggestions
Calc II Suggestions
I skipped college albebra and trigonometry and went straight into Calculus. It was fairly easy, I had to learn trig as I went, but I got an A.
I'm now in Calc II, using a different book through a different school (A university rather than a campus) and I'm starting to have troubles.
Is there a book or a site or a clever system I can study that will broaden my trig understanding? I've considered just buying a trig text book from the campus bookstore.
I've studied the unit circle a lot and played with it on my own, and I have friend that has developed an awesome diagram for multiplication and addition of trig functions, but I assume working through problems is the best thing I can do, but these books are so &%*@&$ expen$ive
the best way is to realize that trig is a special case of the exponential fuinction studied in calculus, and use that to shortcut learning trig.
I myself skipped trig in high school and never learned the usual trig until i had to teach it. the main pooint is that e^(ix) = cos(x) +isin(x), where e^z is defiend by the powers eries e^z = 1 + z + z^2/2! + z^3/3! + z^4/4! +...... for any complex number z.
Calc 2 is tougher than Calc 1, especially in how you apply trig.... just wait for integration methods....trig plays a major role.
I took trig in high school and did not take it seriously so when I got to college and got to calc 2 it had been about 3 years since I took my have effort trig class. I basically had to take a crash course in trig and muscle my way through. I found that the amount of trig in Calc 2 was sufficient for me to become good enough at it, and I got better as I went along. Sure, I was lost some times and I had to take a few more minutes to figure something out at first, but by the final, I knew what identities to use and how to use them. |
The course will cover selected material in Chapters 1
through 16 of the text.The course
introduces skills and concepts which are needed to use the computer in
scientific and engineering work.
Topics include design and analysis of algorithms, methods and techniques of
scientific computation, and the organization of software.
CLASS POLICY
Student Learning Outcomes:Upon completion of this course, the student should be able to:
to understand fundamental programming concepts,
to apply concepts of problem solving to engineering
problems,
to be able to design and implement simple computer
programs in MATLAB to solve engineering problems,
to understand how to approach other computer
languages,
About Matlab:Matlab is an excellent
first language for engineers.It is
an interpreted language that provides students immediate feedback when a program
is run.It is an ideal environment
for ordinary engineering computation.The course is conducted from the Matlab programming environment."Matlab" is a registered trademark of The MathWorks, Inc.Students who wish to work on their Matlab assignments on their own
computers are required to purchase a Matlab license.
For students not wishing to purchase the
license, DSC provides access to computers equipped with Matlab licenses through
a licensing arrangement with The MathWorks, Inc.
Tests:Three chapter tests
will be given over the term along with a comprehensive final exam.The average of the three tests will represent
60% of the class grade.
Test 1Chapter 1 to Chapter 7
Test 2Chapter 8 to Chapter 12
Test 3Chapter 13 to Chapter 16
Homework / Labs:Homework will be assigned daily and discussed the following class
period. Assignments will allow students
to apply the material covered in lecture to programming problems.Labs will be self-paced exercises in assorted computer-related topics.Specified problems will turned in for a grade.The homework average represents
25% of the class grade. .(Late
assignments will be penalized 5 points for each calendar day they are late)
Final Exam:The Final Exam is a comprehensive exam that will represent
15% of the class grade.
The Final Exam is currently scheduled for:Wednesday, May 1, 2013 (10:30 a.m.
-12:30 p.m.)
Final Grades:The course grade will be based on the comprehensive
final exam (15%), homework/labs average (25%), and class test average (60%).
Grade AverageCourse Grade
90 - 100A
80 - 89B
70 - 79C
60 - 69D
Below 60F
OTHER INFORMATION
Attendance:If you are absent from a class, you are responsible for all
material covered and all announcements made.No make-up tests will be given for any reason.If you know that you will be out on a test date due to prior obligations,
then you may make an appointment to take the test early.If you are absent from a test, the final exam will replace your missed
test score (final exam replaces lowest test score).Only one missed test can be replaced; a second or third missed test will
be given a grade of zero.
Honesty:Every student is expected and required to do his or her own work
in this course and all other courses at Dalton State College.Any instances of cheating will result in a minimum penalty of dismissal
from the course with a grade of "F".
Visitors:
College policy allows
only those students registered for a class to attend that class. You may not
bring guests or children to class.
Etiquette:
Talking in class should be
restricted solely to questions and comments pertaining to the material at hand
and intended for me and the whole class to hear.Using cellular phones, pagers, CD players, radios, and/or similar devices
is prohibited in the classroom.
DROP / WITHDRAWAL
The last day to drop this class without penalty is
Thursday, March 21, 2013. You will be
assigned a grade of W.After this date, withdrawal without penalty is permitted only in cases of
extreme hardship as determined by the Vice President for Academic Affairs;
otherwise a grade of WF will be issued.The
proper form for withdrawing from all classes at the college after the official
drop/add period but before the published withdrawal date is the Schedule
Adjustment Form. All students must meet
with a staff member at the Office of Academic Resources in the Pope Student
Center to initiate the withdrawal process.After meeting with the staff member, students will then finalize the
withdrawal process in the Enrollment Services Office.Students who fail to complete the official drop/withdrawal procedure will
receive the grade of F.Withdrawal from class is a student responsibility.The grade of W counts as hours attempted for the purposes of financial aid.
WORKFORCE DEVELOPMENT
If a student receiving aid
administered by the DSC Workforce Development Department drops this class or
completely withdraws from the College, the Schedule Adjustment Form must be
taken to the Workforce Development Office located in Room 214 of the Technical
Education Building. The Office is open on the following schedule:
Students with disabilities or special needs are encouraged
to contact Disability Support Services in Academic Resources.In order to make an appointment to obtain information on the process of
qualifying for accommodations, the
student must contact the Coordinator
of Disability Support Services.
If the college is closed for inclement weather or other
conditions, please consult the course calendar and complete the assigned
sections.If possible, check
GeorgiaView or your email for additional assignments, activities, and due dates.I will load a PowerPoint of the missed lecture in GeorgiaView, and I will
be able to answer questions through email.Compensatory make-up days may be required if the total number of days
lost exceeds the equivalent of one week of class time. |
Temporarily out of stock
Mathematics, a Simple View
Mathematics, A Simple View is a book that covers the fundamentals of Algebra. Except for one, chapter, the book begins each chapter with either an every-day scenario or an illustration. It fuels a grounded understanding, and application thereof, of algebraic problems with their solutions.The book is written for persons age 14 years and upwards.
Dionne Barrett is a graduate of the University of the West Indies. After working with the Jamaican Ministry of Finance and Planning, in 1998 she entered in the Jamaican classroom where she taught Mathematics for a period of about nine and a half years.She began writing Mathematics, A Simple View in 2002, after being encouraged by a former student, who pointed out the value of her math notes. This she completed in 2007.By nature, Dionne is expressive and thinks that Mathematics, although having a language of its own, need not be as complicated as some texts have it. As such, she has sought to make it as reader friendly as possible.
List price:
$53.00
Edition:
N/A
Publisher:
CreateSpace Independent Publishing Platform
Binding:
Trade Paper
Pages:
620
Size:
8.25" wide x 10.75" long x 1 |
textbook presents a number of the most important numerical methods for finding eigenvalues and eigenvectors of matrices. The authors discuss the central ideas underlying the different algorithms and introduce the theoretical concepts required to analyze their behaviour. Several programming examples allow the reader to experience the behaviour of the different algorithms first-hand. The book addresses students and lecturers of mathematics and engineering who are interested in the fundamental ideas of modern numerical methods and want to learn how to apply and extend these ideas to solve new problems. |
Age-Specific Population Models
Allen E. Martin
Abstract
An age-specific population model is built, based on Fibonacci's rabbit problem. The model is examined using spreadsheets, matrices, iteration and exponential regression. The investigations are based on the assumption that students will have a graphing calculator and/or spreadsheet available to them. The suggested investigations encourage students to use a variety of representations and seek links between them. This module might be used as a lead-in to a discussion of Leslie models for population growth, or as an enrichment project after a discussion of exponential regression.
In Liber Abaci, Leonardo of Pisa (Fibonacci, ca. 1202) proposed one of the earliest mathematical models for population growth. The problem situation stated below is a reworking of Fibonacci's original problem which generates an introductory age-specific population model.
Imagine that we start with one pair of rabbits (one female and one male). After N days, this pair matures to reproductive age and immediately produces a new pair. After N more days, the first pair again produce offspring. Thus, each pair of rabbits will reproduce two times during their lifetime (exactly one pair immediately at the start of each new stage, where "pair" always means one female and one male), at intervals separated by N days, and each new pair of rabbits will go on in a similar fashion.
The problem statement suggests that the rabbit population can be broken down into three groups: "newly born", "young adults" and "mature adults". Each pair of newly born rabbits survives to become young adults and to produce one new pair of offspring at this stage. Each pair of young adults survives to become mature adults and to produce another pair of offspring. Finally, each pair of mature adults moves on to "rabbit heaven"; no survival is allowed after stage 3.
This process of moving through the age-structure and the patterns that emerge can be represented several ways:
with diagrams, to break down and understand the dynamics of the problem
with spreadsheets, to capture patterns and create graphs
with matrices, to emphasize the age-structure and make it more explicit
with recursive formulas (iteration), to capture the dynamics algebraically for further analysis
Analysis by Diagram
The first step in understanding the model is to find a way to "make sense" of the problem situation. A chart or diagram like the one shown below is helpful. The columns display the age structure for each of the first 6 time steps. The rows show the first 6 generations.
Diagram Breakdown of the Rabbit Population
Generation
1
NB
YA
MA
2
NB
YA
MA
NB
YA
MA
3
NB
YA
MA
NB
YA
MA
NB
YA
MA
4
NB
YA
MA
NB
YA
NB
YA
NB
YA
NB
YA
5
NB
YA
NB
NB
NB
NB
NB
NB
NB
6
NB
Time Step
1
2
3
4
5
6
Newly Born
1
1
2
3
5
8
Young Adult
0
1
1
2
3
5
Mature Adult
0
0
1
1
2
3
Total
1
2
4
6
10
16
This diagram captures many of the key aspects of the growth process of this rabbit population. Viewing the chart by columns, we can see the age-specific breakdown for each time-step. For example, in the 4th column we see that there are 3 newly born, 2 young adults and 1 mature adult. Viewing the chart by rows, we see the progression of the pairs born in a given generation as they move through the age-specific categories for the rabbit population. For example, the two pairs born in the 3rd generation become young adults in the next column, contributing 2 pairs of newly born to the 4th generation below them; they then survive to produce one last time, contributing to the 5th generation.
Analysis by Spreadsheet
The information contained in the diagram can then be summarized in a spreadsheet like the one shown below.
Time
Newly
Young
Mature
Total of
Step
Born
Adults
Adults
Rabbits
1
1
0
0
1
2
1
1
0
2
3
2
1
1
4
4
3
2
1
6
5
5
3
2
10
6
8
5
3
16
7
13
8
5
26
8
21
13
8
42
9
34
21
13
68
10
55
34
21
110
Once the spreadsheet has been created, we can view large amounts of data conveniently, include the data in reports, and easily create graphs. Also, we can vary the assumptions of the model and explore variations of the problem situation quickly.
Investigation #1
Starting with 1 pair of "newly born" rabbits, suppose that each pair of rabbits survives through 4 time steps, instead of three.
(a) Create a "diagram analysis" to break this problem situation down
(b) Create a spreadsheet that shows the data for each of the 4 age categories
(c) Create a graph showing Newly Born vs. Time Step
Investigation #2
Suppose that each pair of rabbits survives through 3 time steps (as in the original setup), but that each pair of young adults has 2 pairs of newborns. Also suppose that mature adults have only 1 pair each.
(a) Create a table showing each age category and the total number of rabbits for time steps = 1, 2, 3, ... , 10
(b) Create a graph showing Newly Born vs. Time Step
Analysis with Matrices
Let's return to the original problem situation. For any given time step, the population can be conveniently broken down into its age-specific groups with matrix notation.
So the information in the diagram and spreadsheet can be expressed as follows:
Step123456 n
Structure
Now, as the population moves from one time step to the next, we see that
Bold ---> Ynew , Yold ---> Mnew, and then Ynew + Mnew ---> Bnew
This transformation can be accomplished by matrix multiplication!
This can be expressed in a more compact form:
where T is the transition matrix
and Pold & Pnew are the population matrices.
This multiplication can be accomplished on a calculator with "Answer-Key" iteration. First enter the transition matrix into matrix [A], then the initial population matrix into matrix [B]. Next call matrix [B] and press <enter>. Then call matrix [A] and multiply by ANS. Finally, press <enter>, <enter>, <enter>, ... to get the 2nd, 3rd, 4th, ... generations.
Analysis by Iteration
As can be seen in both the diagram and the spreadsheet, the values of each age group can be determined from previous values. These patterns can be expressed iteratively.
Let = # of newly born in the nth time step
= # of young adults in the nth time step
= # of mature adults in the nth time step
then, moving from one time-step to the next, we can see that
,
and then
Note: It follows that
Since all three age groups have the characteristic Fibonacci-like pattern
(b) Extend this table further out to the right (by letting n = 10, 20, 30, ...); then describe what happens to the values of .
(c) Sketch a graph of vs. n.
In the third investigation, we find that grows exponentially:
for n = 1, 2, 3, ..., 10
and, for n = 1, 2, 3, ..., 20
.
The fourth investigation reinforces this, since the ratio gets very close to 1.618 as n gets large; hence .
In fact, if we assume that and apply this to the iteration , we get
The roots of this equation are
Key Observation:Notice the striking similarity between the base of the exponential, the limiting value of the ratio and one of the roots of this "characteristic polynomial". What is going on here? The next two investigations will explore this similarity further.
Investigation #5 --
In investigation #1, we found that the "newly born" followed the pattern shown above. Using this ...
(a) Determine the characteristic polynomial for this iteration, then graph this polynomial on your calculator. How many real roots does it have? Approximate any real roots you find. |
The aim of this book is to present a modern treatment of the theory of theta functions in the context of algebraic geometry. The novelty of its approach lies in the systematic use of the Fourier-Mukai transform. more...
Accessible Mathematics is Steven Leinwand?s latest important book for math teachers. He focuses on the crucial issue of classroom instruction. He scours the research and visits highly effective classrooms for practical examples of small adjustments to teaching that lead to deeper student learning in math. Some of his 10 classroom-tested teaching shifts... more...
impressive volume is dedicated to Mel Nathanson, a leading authoritative expert for several decades in the area of combinatorial and additive number theory. For several decades, Mel Nathanson's seminal ideas and results in combinatorial and additive number theory have influenced graduate students and researchers alike. The invited survey articles... more...
Markov Chain Monte Carlo (MCMC) methods are now an indispensable tool in scientific computing. This book discusses recent developments of MCMC methods with an emphasis on those making use of past sample information during simulations. The application examples are drawn from diverse fields such as bioinformatics, machine learning, social science, combinatorial... more...
The selected papers in this volume cover all the most important areas of ring theory and module theory such as classical ring theory, representation theory, the theory of quantum groups, the theory of Hopf algebras, the theory of Lie algebras and Abelian group theory. The review articles, written by specialists, provide an excellent overview of the... more...
Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important... more...
Features contributions that are focused on significant aspects of current numerical methods and computational mathematics. This book carries chapters that advanced methods and various variations on known techniques that can solve difficult scientific problems efficiently. more...
Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology... more... |
Scope and form:
Duration of Course:
F2B
The exam date is only used to specify the deadline for the report (c.f. evaluation)
Type of assessment:
Evaluation of exercises/reports Five homework sets and three quizzes during the semester, and a final report on a project exercise. The homework and the quizzes counts together for 60% of the grade, and the report counts 40% of the grade.
Aid:
All Aid
Evaluation:
7 step scale, internal examiner
Qualified Prerequisites:
General course objectives:
The aim of this course is to provide the students with basic tools and competences regarding the analysis and applications of curves and surfaces in 3D. The main idea of the course is very well described by the following exerpt from the cover of the textbook: "Curves and surfaces are objects that everyone can see, and many questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques." One integral part of the course is to apply computer experiments with Maple in eachone of the three steps: To ask natural geometric questions, to formulate them in precise mathematical terms, and to answer them using techniques from calculus. The course also aims to give the students a firm background for further studies in the manifold engineering applications of differential geometric tools and concepts.
Learning objectives:
A student who has met the objectives of the course will be able to:
Calculate the curvature, the torsion, and the Frenet-Serret basis for a given space curve.
Apply the first and second fundamental form to analyze curves on surfaces in space.
Recognize isometries and conformal maps between simple surfaces.
Determine the principal curvatures and principal directions at every point of a given surface.
Calculate the Gauss curvature and the mean curvature at every point of a given surface.
Explain the invariant geometric significance of the Gauss curvature.
Explain the connection between the second fundamental form, the Weingarten map and the principal curvatures and directions.
Explain the connection between the total curvature, the normal curvature, and the geodesic curvature of a curve on a given surface.
Recognize geodesic curves from data about the normal curvature and the total curvature of the curves.
Apply the general surface theory to surfaces of revolution and to ruled surfaces.
Apply the Gauss-Bonnet theorem to estimate the Euler characteristic of a given surface.
Apply the general theory to a simple geometric problem and present the solution in the form of a report.
Content:
Curves and surfaces in 3D - with particular focus on metric and curvature properties. How to find the shortest path on a surface. How to bend a surface. How to calculate the number of holes of a compact surface. Individually chosen applications of differential geometry which span a diversity of possibilities including roler coaster constructions, geographic map projections, relativity (special or general), protein geometry, to mention but a few. The specific list of contents includes: Curves with constant width, the Frenet-Serret 'apparatus' for curves in 3D, first and second fundamental forms for surfaces, Gaussian curvature and mean curvature, equiareal maps, isometries, surfaces of constant curvature, geodesics, fundamental results of Gauss, Codazzi-Mainardi, and Gauss-Bonnet.
Course literature:
Andrew Pressley: Elementary Differential Geometry, Springer, 2001 |
Elementary Linear Algebra
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study.
The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, exploring a comprehensive range of topics. |
Seventh-grade extended math completes the remaining seventh grade and all of the eight grade Standards of Learning set by the State of Virginia, and in some cases goes beyond state standards. Students in this course will take the Grade 8 Mathematics SOL exam in the spring.
The seventh-grade standards place emphasis on solving problems involving consumer applications, using proportional reasoning, and gaining proficiency in computations with integers. The students will gain an understanding of the properties of real numbers, solve one-step linear equations and inequalities, and use data analysis techniques to make inferences, conjectures, and predictions. Two- and three-dimensional representations, graphing transformations in the coordinate plane, and probability will be extended.
The eighth-grade standards contain both content that reviews or extends concepts and skills learned in previous grades and new content that prepares students for more abstract concepts in algebra and geometry. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and use proprotions to solve a variety of problems. New concepts include solving two-step equations and inequalities, graphing linear equations, visualizing three diminsional shapes represented in two-dimensional drawings, applying transformations to geometric shapes in the coordinate plane, and using matrices to organize and interpret data. Students will verify and apply the Pythagorean Theorem and represent relations and functions using tables, graphs, and rules.
For a summary of the Standards of Learnings taught in the seventh-grade extended math see the parent-brochure below.
Documents
PWCS Extended Grade 7 Curriculum Guide (Word) - This document contains the pacing guide and curriculum guide to be used beginning with school year 2012-2013 for Grade 7 Extended Mathematics. The pacing guide has been updated. Updated August 2012.
Extended Grade 7 Curriculum Map (Word) - This document is an abbreviated form of the curriculum guide. It only contains the objectives and essential knowledge and skills. It is in a table format with an extra column. It may be used to create blueprints for unit assessments, making notes for units and lesson plans etc.
SOL Electronic Practice Items and Tools Practice - contains a sample of practice items for each standards of learning test and a companion guide to assist in stepping through the sample problems and learning the online tools for the technically enhanced questions. There is also a separate application for just practicing with the online tools |
About the book
Mathematics for the International Student: Mathematics HL has been written to reflect the
syllabus for the two-year IB Diploma Mathematics HL course. It is not our intention to define the
course. Teachers are encouraged to use other resources. We have developed the book independently
of the International Baccalaureate Organization (IBO) in consultation with many experienced
teachers of IB Mathematics. The text is not endorsed by the IBO.
This second edition builds on the strengths of the first edition. Many excellent suggestions were
received from teachers around the world and these are reflected in the changes. In some cases
sections have been consolidated to allow for greater efficiency. Changes have also been made in
response to the introduction of a calculator-free examination paper. A large number of questions,
including some to challenge even the best students, have been added. In particular, the final chapter
contains over 200 miscellaneous questions, some of which require the use of a graphics calculator.
These questions have been included to provide more difficult challenges for students and to give
them experience at working with problems that may or may not require the use of a graphics
calculator.
The combination of textbook and interactive Student CD will foster the mathematical development
of students in a stimulating way. Frequent use of the interactive features on the CD is certain to
nurture a much deeper understanding and appreciation of mathematical concepts.
The book contains many problems from the basic to the advanced, to cater for a wide range of
student abilities and interests. While some of the exercises are simply designed to build skills,
every effort has been made to contextualise problems, so that students can see everyday uses and
practical applications of the mathematics they are studying, and appreciate the universality of
mathematics.
Emphasis is placed on the gradual development of concepts with appropriate worked examples, but
we have also provided extension material for those who wish to go beyond the scope of the
syllabus. Some proofs have been included for completeness and interest although they will not be
examined.
For students who may not have a good understanding of the necessary background knowledge for
this course, we have provided printable pages of information, examples, exercises and answers on
the Student CD. To access these pages, simply click on the 'Background knowledge' icons when
running the CD.
It is not our intention that each chapter be worked through in full. Time constraints will not allow
for this. Teachers must select exercises carefully, according to the abilities and prior knowledge of
their students, to make the most efficient use of time and give as thorough coverage of work as
possible.
Investigations throughout the book will add to the discovery aspect of the course and enhance
student understanding and learning. Many Investigations could be developed into portfolio
assignments. Teachers should follow the guidelines for portfolio assignments to ensure they set
acceptable portfolio pieces for their students that meet the requirement criteria for the portfolios.
Review sets appear at the end of each chapter and a suggested order for teaching the two-year
course is given at the end of this Foreword.
The extensive use of graphics calculators and computer packages throughout the book enables
students to realise the importance, application and appropriate use of technology. No single aspect
of technology has been favoured. It is as important that students work with a pen and paper as it is
that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on
computer.
The interactive features of the CD allow immediate access to our own specially designed geometry
packages, graphing packages and more. Teachers are provided with a quick and easy way to
demonstrate concepts, and students can discover for themselves and re-visit when necessary.
Instructions appropriate to each graphic calculator problem are on the CD and can be printed for students.
These instructions are written for Texas Instruments and Casio calculators.
In this changing world of mathematics education, we believe that the contextual approach shown in this
book, with the associated use of technology, will enhance the students' understanding, knowledge and
appreciation of mathematics, and its universal application.
Using the interactive student CD
The interactive CD is ideal for independent study. Frequent use will nurture
a deeper understanding of Mathematics. Students can revisit concepts taught in
class and undertake their own revision and practice. The CD also has the text of
the book, allowing students to leave the textbook at school and keep the CD at
home.
The icon denotes an Interactive Link on the CD. Simply 'click' the
icon to access a range of interactive features:
spreadsheets
video clips
graphing and geometry software
graphics calculator instructions
computer demonstrations and simulations
background knowledge (as printable pages)
For a complete list of all the active links on the Mathematics HL CORE second edition CD,
click here.
For those who want to make sure they have the prerequisite levels of understanding
for this course, printable pages of background informations, examples, exercises and
answers and provided on the CD. Click the 'Background knowledge' icon on
pages 12 and 248.
Graphics calculators: Instructions for using graphics calculators are also given on
the CD and can be printed. Instructions are given for Texas Instruments and Casio calculators.
Click on the relevant icon (TI or C) to access the instructions for the other type of
calculator.
Note on accuracy
Students are reminded that in assessment tasks, including examination papers, unless
otherwise stated in the question, all numerical answers must be given exactly or to
three significant figures.
HL and SL combined classes
HL Options
This is a companion to the Mathematics HL (Core) textbook. It offers coverage
of each of the following options:
Topic 8 – Statistics and probability
Topic 9 – Sets, relations and groups
Topic 10 – Series and differential equations
Topic 11 – Discrete mathematics
In addition, coverage of the Geometry option for students undertaking the IB Diploma
course Further Mathematics is presented on the CD that accompanies the
HL Options book.
Supplementary books
A separated book of WORKED SOLUTIONS give the fully worked solutions for every question
(discussions, investigations and projects excepted) in each chapter of the
Mathematics HL (Core) textbook. The HL (CORE) EXAMINATION PREPARATION & PRACTICE
GUIDE offers additional questions and practice exams to help students prepare for the
Mathematics HL examination. For more information email
[email protected]. |
Math 8 Textbook Website: Applications and Connections (Course 3) (The Parent and Student Guide has worksheets that you can print out for each chapter. These one-page worksheets have a very brief, but clear explanation of each lesson and some examples for practice with answers at the bottom of the page.) |
Famous Problems of Geometry and How to Solve Them by Benjamin Bold Delve into the development of modern mathematics and match wits with Euclid, Newton, Descartes, and others. Each chapter explores an individual type of challenge, with commentary and practice problems. SolutionsA Concept of Limits by Donald W. Hight An exploration of conceptual foundations and the practical applications of limits in mathematics, this text offers a concise introduction to the theoretical study of calculus. Many exercises with solutions. 1966 editionProduct Description:
more than 80 drawings. 1963 |
Careers in math: The
Funworks site describes many career
fields, and includes a category of careers that use math, useful for when you have to write a "how
people use math in real life" sort of paper.
Common Errors in College Math: Professor Eric Schechter has compiled an extensive list of
the errors most commonly seen in college math courses. While some of his illustrations are specific
to calculus, any high-school or college algebra student would benefit from reviewing his warnings
and recommendations.
Culturally Situated Design Tools:
If you need to do a "math in other cultures" report, this
site might be helpful. Their connections between "culture" and math seem fairly strained
(do you really think ancient Africans were "engaging" in transformational geometry
when they were braiding their kids hair?), but their articles should give you what your teacher
is wanting to hear.
Curious and Useful Math:
Clay Ford has developed a "curious" site
which covers taking square roots by hand, doing mental math, and playing math tricks, among other
things.
Cut-the-Knot: A fascinating site,
you will find topics that go right over your head (mine, too) right next to topics that usefully
answer annoying take-home-project homework questions. Many topics have Javascript illustrations.
Earliest Uses of Math Terms and Symbols: These pages are great for finding the first known use of the division
symbol, the fraction bar, the term "x-intercept", or other mathematical terms or symbols.
Finding your Way Around:
MathBits.com has an extensive listing of instructions
for using your Texas Instruments TI-83 or TI-84 calculator for various tasks.
Graphing Calculator
Help: The Prentice-Hall publishing company has extensive examples
for some Texas Instruments graphing calculator models, along with the Casio FX2, the Sharp
EL9600C, the CFX-9850, and HP48G.
MailWasher Pro:
Okay, this isn't math-related, but I love this e-mail utility. The MailWasher (MW) program
isn't really for catching or preventing spam, though it does have tools that can help in that regard.
The great thing about MW is that it allows you to delete the garbage e-mails straight off
your ISP's mail server; the junk never touches your computer. You can try the program before
buying; it took me about twenty minutes of using MW to decide that it was worth the price.
Math and Music:
If you need to write a paper on "applications of math", the research posted by
Professor Hall might be a good place to start. Note: Her resources are largely PDF files, so make
sure you've got Acrobat Reader installed.
Math in Daily Life: If you need to write a paper on "How Math is Useful
in Everyday Life", this
is a good place to start.
Mathematical Fiction:
This categorized and searchable index of fictional
works relating to mathematics is categorized according to medium (comic books, movies, etc), genre
(horror, sci-fi, etc), motif (aliens, music, insanity, etc), and topic (logic, chaos, etc). You
can also search by title or author, or in chronological order. If you're having to do a report
or presentation on "something mathematical", this could be a great place to start.
Mathematical Moments:
This American Mathematical
Society page lists various disciplines in
which math is used "in real life". The first two links for each topic take you to a descriptive
flyer; the third link takes you to the actual information.
Mathwords:
If you're looking for a definition, try this site. There is an alphabetic
index available along the left-hand side of the page, or you can enter your term into the search
box in the top right corner.
Meracl FontMap: Did you know that the degree symbol " ° ", the
empty-set symbol "Ø", the division symbol "÷", and the plus-minus symbol "±"
are all standard characters, and that you can type them into e-mails and such without having to
resort to special fonts that your recipient might not be able to read? This free
program lets you "see" all these characters in your preferred font, and lets you
paste them into your document. A "must" for those of us who can't read the tiny little
"Character Map" that comes with Windows.
The Nine Digits
Page: If you have one of those "impossible"
puzzle problems that you have to solve using the digits "1" through "9", this site might contain the archived solution.
No Boundaries:
USAToday has created a service allowing the interested student to "explore careers in science, technology, engineering and math" within NASA. The site is designed for
group projects within the classroom, but the information can be adjusted to, say, help in writing
a report on "careers in math".
Practical Uses of Math and Science:
Need to write one of those "how math is used in real life" papers? Check out the PUMAS listing created by NASA. Topics include how to calculate square roots
with a carpenter's square, the mathematical implications of lying, and why clouds don't fall out
of the sky. Click on the title, and then click on "View this Example".
Print Free Graph Paper:
The PDFPad site provides free
graph paper that you can view and print
with the Acrobat Reader browser plug-in.
Stan Brown's Math and Calculator articles: Professor Brown has created a nice collection of tutorials
covering many common tasks, and some not-so-common ones, for classes from algebra through calculus
and statistics. Includes programs you can download and install, step-by-step instructions, illustrations,
and a conversational tone.
TI-83 tutorial: This
highly graphical tutorial covers some basic topics such as setting the window, creating
a table, and doing a regression. A floating red arrow takes you step-by-step through the processes.
Women in Mathematics:
Need to write a paper on minorities, or specifically on women, in mathematics? This might be a
good place to start.
If you have found a particular web
site to be useful for geometry, proofs, trigonometry, linear algebra, or calculus, please let me know. Thank you!
If you think your site should be listed here, please submit
the URL, explaining how you think your free (or free-to-try) products and/or services
would aid algebra students. Listings are added at the webmistress' discretion; listings for "calculators"
and "graphers" are no longer accepted. Sorry. |
Introduction to or graduate courses in Management Science, Quantitative Methods, and Decision Modeling. Introduction to Management Science shows students how to approach decision-making problems in a straightforward, logical way. By focusing on simple, straightforward explanations and examples with step-by-step details of the modeling and solution techniques, this text makes the mathematical topics of Management Science less complex. The tenth edition retains the same readability and accessibility to techniques and applications as the widely-... MOREadopted previous editions, and also includes updated Excel spreadsheets, Excel "Add-ins", and new problems and case studies. Introduction to Management Scienceshows readers how to approach decision-making problems in a straightforward, logical way. Management Science; Linear Programming: Model Formulation and Graphical Solution; Linear Programming: Computer Solution and Sensitivity Analysis; Linear Programming: Modeling Examples; Integer Programming; Transportation, Transshipment, and Assignment Problems; Network Flow Models; Project Management; Multicriteria Decision Making; Nonlinear Programming; Probability and Statistics; Decision Analysis; Queuing Analysis; Simulation; Forecasting; Inventory Management MARKET: This text equips readers with the skills and knowledge they need to solve problems through the use of mathematical models and computer solutions that implement the latest technology. Taylorrs"s objective was to focus on using simple, straightforward explanations and detailed step-by-step examples that readers would find understandable and easy-to-read. |
Develop a rich understanding of the rudiments of algebra in a relaxed and supportive learning environment. This course will help you understand some of the most important algebraic concepts, including orders of operation, units of measurement, scientific notation, algebraic equations, inequalities with one variable, and applications of rational numbers.
Learning Objectives:
Identify types of numbers, practical applications, and history of algebra.
You will need to create a login for your online classroom. Go to Find your course by browsing the catalog or using the search bar. Click the "Enroll Now" button. Select your start date and then create a Username and Password. At the end, you will be asked to click on the "Already Paid" button if you have already paid. You must make an 80 or higher on the final exam (online) to successfully complete the course. You may only take the exam once.
If you are a certified teacher in Georgia and are interested in taking this course for PLUs, please complete the PLU notification form once you have registered. This course grants 2 PLUs upon successful completion.
If you have questions about this course, please contact the online coordinator at 770-499-3355 or [email protected]. |
An investigation of topics including the history of mathematics, number systems, geometry, logic, probability, and statistics. There is an emphasis throughout on problem solving. Recommended for General Education.
For some of you this course might serve to satisfy the math competency requirement, for others this will be just one of the mathematics courses required by your major/minor program. In any case, the main goal of this course is to expose you to a variety of areas of mathematics and thus give you an idea of the importance of mathematics in today's world and a multitude of ways it is being used in practice. We will learn some elements of mathematical logic, set theory, geometry, statistics, probability, consumers mathematics, and some basic algebra.
The content and the methods of this course are designed in accordance with general education objectives and the work in this course should help you in developing a number of skills included in the NCTM (National Council of Teachers of Mathematics) `standards'' for mathematics education, and also being among the general education objectives at Viterbo. The main emphasis throughout the course will be on problem solving and developing thinking skills. This includes: writing numbers and performing calculations in various numeration system, solving simple linear equations, exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments, exploring a few major concepts of Euclidean Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem, develop an ability to use deductive reasoning, in the context of the rules of logic and syllogisms, i.e., learn how to make/recognize a valid argument, some basics of probability and statistics ...
Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics. In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics.
Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today's world, learning how to handle simple loans, learning how to reason correctly and make a valid argument), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills.
I encourage you to read the text at: - the Viterbo critical thinking web page
Text
Robert Blitzer, Thinking Mathematically, Prentice-Hall, 2000.
Format
Class sessions will consist of lectures, work in small groups, exams, and individual presentations. I expect students to work out the recommended practice problems and ask for help whenever needed.
Resources
Please do not hesitate to contact me for any question you might have; do not let a feeling such as ``I am lost ...'' to last.
Internet and the blackboard software. There is a lot of material on my web page. I will use the Blackboard to communicate with you, so please check your e-mail regularly. I would also like to encourage you to explore, and use numerous capabilities of that (Blackboard) software.
Learning center.
Library. Note that a video set that covers your textbook exists.
Grading
The following grading scale applies to individual exams, and to the overall grade as well:
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.
The following exceptions to that scale are possible:
An A on the final exam (more than points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, ....
An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, ....
If one is failing the course by the end of the semester, but has over average on exams, and earns at least points on the final, he/she can get a D for the final grade.
If one is passing the course by the time of the final exam, but earns less than points (a score less than), that will result in an F for the final grade.
Assignments
Recommended practice: First, middle and the last problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical Thinking Exercises.
These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems.
Two essays, points each:
Autobiography: Introduce yourself to me in a 2-3 pages essay. State your name, and where (city/state) you are coming from. The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course?
This assignment is due Friday, January 18.
World without mathematics: another 2-3 pages 20 points essay.
Try to imagine, and describe, a world without mathematics.
Due: Friday, January 25.
Homework
At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the ``test''is worth 1 point.
Exams
There will be three in-class exams, worth points each. An exam will typically cover three chapters worth of material. The exams will be closed notes, closed book. However, a calculator and a formula sheet (but not any worked out problem) is allowed. Before each exam, I will give you a take-home practice exam, which will be very much like the actual exam coming. I will grade (25 points) the first one of those, i.e., the ``Exam 1 - Practice'', but not the others. I will also allow a makeup (up to 50%) of the lost credit for the exam 2. This makeup will be oral, and will apply to those under points on the test, and is to be done within two weeks after the exam.
Final Exam
Final exam is a 2-hour, cumulative exam, and is worth points.
Portfolio
It should consist of 5 problems, but no two problems should be of the same type (from the same section).
Format: You state a problem, write a complete/correct solution to it, and then write a paragraph (or more) explaining why did you choose that particular problem, what did you learn from it, etc..
The portfolio will be worth points.
The problems you choose for the portfolio should illustrate the progress in learning mathematics, the change of the perception (if any) of what mathematics is about, the change (if any) in your perception about your abilities to do mathematics.
In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet.
Typically, the explanations you will find on the Internet are a bit sketchy. So part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then to clearly explain that proof to your class mates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help.
You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have.
The presentation will be worth points. In addition to that, one certain problem for one of the exams, and for the final exam is going to be:
State and prove the Pythagorean Theorem.
Group Labs: At a number of points during the course you will be working on a ``lab'' in small groups. Even though you will be working in a group of three or four people, each person should turn in a paper. It is important that each person contributes their input into these labs. However, I expect you to write the turn-in paper all by yourself.
Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and Wayne Wojciechowski in Murphy Center Room 320 (796-3085) within ten days to discuss your accommodation needs. |
Upcoming Events
Courses
Courses offered in the past four years. ▲indicates offered in the current term ▹indicates offered in the upcoming term[s]
MATH 0100 - A World of Mathematics
▲
A World of Mathematics
How long will oil last? What is the fairest voting system? How can we harvest food and other resources sustainably? To explore such real-world questions we will study a variety of mathematical ideas and methods, including modeling, logical analysis, discrete dynamical systems, and elementary statistics. This is an alternative first mathematics course for students not pursuing the calculus sequence in their first semester. The only prerequisite is an interest in exploring contemporary issues using the mathematics that lies within those issues. (This course is not open to students who have had a prior course in calculus or statistics.) 3 hrs lect./disc.
MATH 0109 - Mathematics for Teachers
▹
Mathematics for Teachers
What mathematical knowledge should elementary and secondary teachers have in the 21st century? Participants in this course will strengthen and deepen their own mathematical understanding in a student-centered workshop setting. We will investigate the number system, operations, algebraic thinking, measurement, data, and functions, and consider the attributes of quantitative literacy. We will also study recent research that describes specialized mathematical content knowledge for teaching. (Not open to students who have taken MATH/EDST 1005. Students looking for a course in elementary school teaching methods should consider EDST 0315 instead.)▲▹ 0122 - Calculus II
▲▹
Calculus II
A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations. (MATH 0121 or by waiver) 4 hrs. lect./disc.
MATH 0190 - Math Proof: Art and Argument
▹
Mathematical Proof: Art and Argument
Mathematical proof is the language of mathematics. As preparation for upper-level coursework, this course will give students an opportunity to build a strong foundation in reading, writing, and analyzing mathematical argument. Course topics will include an introduction to mathematical logic, standard proof structures and methods, set theory, and elementary number theory. Additional topics will preview ideas and methods from more advanced courses. We will also explore important historical examples of proofs, both ancient and modern. The driving force behind this course will be mathematical expression with a primary focus on argumentation and the creative process. (MATH 0122 or MATH 0200) 3 hrs. lect.
MATH 0200 - Linear Algebra
▲▹ 0217 - Elements of Math Bio & Ecol
▲
Elements of Mathematical Biology and Ecology
Mathematical modeling has become an essential tool in biology and ecology. In this course we will investigate several fundamental biological and ecological models. We will learn how to analyze existing models and how to construct new models. We will develop ecological and evolutionary models that describe how biological systems change over time. Models for population growth, predator-prey interactions, competing species, the spread of infectious disease, and molecular evolution will be studied. Students will be introduced to differential and difference equations, multivariable calculus, and linear and non-linear dynamical systems. (MATH 0121 or by waiver)
MATH 0223 - Multivariable Calculus
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Multivariable Calculus
The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 or MATH 0200 or by waiver) 3 hrs. lect./disc.
MATH 0247 - Graph Theory
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Graph Theory
A graph (or network) is a useful mathematical model when studying a set of discrete objects and the relationships among them. We often represent an object with a vertex (node) and a relation between a pair with an edge (line). With the graph in hand, we then ask questions, such as: Is it connected? Can one traverse each edge precisely once and return to a starting vertex? For a fixed k/, is it possible to "color" the vertices using /k colors so that no two vertices that share an edge receive the same color? More formally, we study the following topics: trees, distance, degree sequences, matchings, connectivity, coloring, and planarity. Proof writing is emphasized. (MATH 0122 or by waiver) 3 hrs. lect./disc.
MATH 0250 - Ethnomathematics
Ethnomathematics: A Multicultural View of Mathematical Ideas and Methods*
What are the cultural roots of the mathematics we study and use today? Even though it has been developed by individuals from widely varying cultural contexts, we take the verity, consistency, and universality of mathematics for granted. How does the western tradition stand in comparison to the mathematics developed by indigenous societies, labor communities, religious traditions, and other groups that can be studied ethnographically? By examining the cultural influences on people and the mathematics they practice, we shall deepen our understanding of mathematics and its relationship to society. 3 hrs. lect/disc.
MATH 0261 - History of Mathematics
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History of Mathematics
This course studies the history of mathematics chronologically beginning with its ancient origins in Babylonian arithmetic and Egyptian geometry. The works of Euclid, Apollonius, and Archimedes and the development of ancient Greek deductive mathematics is covered. The mathematics from China, India, and the Arab world is analyzed and compared. Special emphasis is given to the role of mathematics in the growth and development of science, especially astronomy. European mathematics from the Renaissance through the 19th Century is studied in detail including the development of analytic geometry, calculus, probability, number theory, and modern algebra and analysis. (MATH 0122 or waiver)
MATH 0311 - Statistics
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Statistics
An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines. (MATH 0310) 3 hrs. lect./disc.
MATH 0315 - Mathematical Models
Mathematical Models in the Social and Life Sciences
An introduction to the role of mathematics as a modeling tool and an examination of some mathematical models of proven usefulness in problems arising in the social and life sciences. Topics will be selected from the following: axiom systems as used in model building, optimization techniques, linear and integer programming, theory of games, systems of differential equations, computer simulation, stochastic process. Specific models in political science, ecology, sociology, anthropology, psychology, and economics will be explored. (MATH 0200 or waiver) 3 hrs. lect./disc.
MATH 0318 - Operations Research
▲ 0335 - Differential Geometry
Differential Geometry
This course will be an introduction to the concepts of differential geometry. For curves in space, we will discuss arclength parameterizations, Frenet formulas, curvature, and torsion. On surfaces, we will explore the Gauss map, the shape operator, and various types of curvature. We will apply our knowledge to understand geodesics, metrics, and isometries of general geometric spaces. If time permits, we will consider topics such as minimal surfaces, constant curvature spaces, and the Gauss-Bonnet theorem. (MATH 0200 and MATH 0223) 3 hr. lect./disc.
MATH 0345 - Combinatorics
Combinatorics
Combinatorics is the "art of counting." Given a finite set of objects and a set of rules placed upon these objects, we will ask two questions. Does there exist an arrangement of the objects satisfying the rules? If so, how many are there? These are the questions of existence and enumeration. As such, we will study the following combinatorial objects and counting techniques: permutations, combinations, the generalized pigeonhole principle, binomial coefficients, the principle of inclusion-exclusion, recurrence relations, and some basic combinatorial designs. (MATH 0200 or by waiver) 3 hrs. lect./disc.
MATH 0402 - Topics In Algebra
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Topics in Algebra
A further study of topics from MATH 0302. These may include field theory, algebraic extension fields, Galois theory, solvability of polynomial equations by radicals, finite fields, elementary algebraic number theory, solution of the classic geometric construction problems, or the classical groups. (MATH 0302 or by waiver) 3 hrs. lect./disc. 0423 - Topics in Analysis
Topics in Analysis
In this course we will study advanced topics in real analysis, starting from the fundamentals established in MA401. Topics may include: basic measure theory; Lebesgue measure on Euclidean space; the Lebesgue integral; total variation and absolute continuity; basic functional analysis; fractal measures. (MATH 0323 or by waiver) 3 hrs. lect./disc.
MATH 0432 - Elementary Topology
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Elementary Topology
An introduction to the concepts of topology. Theory of sets, general topological spaces, topology of the real line, continuous functions and homomorphisms, compactness, connectedness, metric spaces, selected topics from the topology of Euclidean spaces including the Jordan curve theorem. (MATH 0122 or by waiver▲▹
MATH 1001 - The Game of Go
The Game of Go
Go is the most ancient of all East Asian board games and the most challenging in play. It combines the logic of mathematics with the aesthetic appeal of music and art. Despite this, it is a simple game to learn and is enjoyed by approximately 35 million enthusiasts the world over including over 1000 dedicated professionals. The course will involve playing, recording, analyzing, and critiquing our games and learning about its history and the cultures in which it flourishes. We will also read and write about various related Japanese arts and traditions. (Not open to students who have taken FYSE 1175).
MATH 1004 - The Shape of Space
The Shape of Space
We know that the earth we live on is a sphere, but consider the three-dimensional shape of the universe. Does it go on forever, or could it wrap back on itself in some way? In this course we will consider the shape of space. We will learn how topologists and geometers visualize three-dimensional spaces, with a goal of learning about the eight three-dimensional shapes that form the building blocks of all three-dimensional spaces. In the process, we will learn about the celebrated Poincare Conjecture. The ideas we encounter will be deep, but we will study them in a hands-on way.
MATH 1005 - Mathematics for Teachers
Mathematics for Teachers
What mathematical knowledge should elementary and secondary teachers have? We will investigate recent research that describes specialized mathematical content knowledge for teaching. Participants in this course will also strengthen and deepen their own mathematical understanding in a student-centered workshop setting. Readings include Knowing and Teaching Elementary Mathematics by Liping Ma as well as readings from the Journal for Research in Mathematics Education. Anyone interested in mathematics education at any level is welcome. (Not open to students who have taken MATH 1003)
MATH 1006 - Heart of Mathematics
The Heart of Mathematics
Wrestling with the infinite, tiling a floor, predicting the shape of space, imagining the fourth dimension, untangling knots, making pictures of chaos, conducting an election, cutting a cake fairly; all of these topics are part of the landscape of mathematics, although they are largely excluded from the calculus-centric way that the subject is traditionally presented. Following the acclaimed text, The Heart of Mathematics, by Ed Burger and Michael Starbird, we will dive headfirst into ideas that reveal the beauty and diverse character of pure mathematics, employing effective modes of reasoning that are useful far beyond the boundaries of the discipline.
MATH 1007 - Combinatorial Gardner
The Combinatorial Gardner
It has been said that the Mathematical Games column written by Martin Gardner for Scientific American turned a generation of children into mathematicians and mathematicians into children. In this course we will read selections from three decades of this column, focusing on those that deal with combinatorics, the "science of counting," and strive to solve the problems and puzzles given. An example problem that illustrates the science of counting is: what is the maximum number of pieces of pancake (or donut or cheesecake) one can obtain via n linear (or planar) cuts? (MATH 0116 or higher; Not open to students who have taken FYSE 1314).
MATH 1038 - Combinatorial Games & Puzzles
Combinatorial Games and Puzzles
Games and puzzles with a combinatorial flair (based on counting and arrangement) have entertained and frustrated people for millennia. Mathematicians have developed new areas of research and discovered non-trivial mathematics upon examining these amusements. Students will play games (including nim, hex, dots-and-boxes, clobber, and Mastermind®) and be presented with puzzles (including instant insanity and mazes) in an attempt to develop strategy and mathematics during play. Basic notions in graph theory, design theory, combinatorics, and combinatorial game theory will be introduced. Despite the jargon, this course will be accessible to all regardless of background.
MATH 1095 - Statistical Computing with R
Statistical Computing with R
This course offers an intensive introduction to the R statistical programming environment. Students will learn to use a modern programming language that incorporates object-oriented programming. Topics will include data frames, the R environment, the graphics system, probability distributions, descriptive statistics, statistical tests and confidence intervals, regression, ANOVA, tables of counts, simulation, and selected topics in statistical programming. (One course in statistics or one course in computer programming). |
Algebra 1 Operations With Word ProblemBy:-Matthew David
Algebra is one of the most basic element of mathematics in which, we switch from basic arithmetic to variables. Here instead of using numbers we use different variables to represent different parameters. Algebra 1 is taught in initial level learning of algebra. They involve with basic terms like addition, subtraction and multiplication of variables, they also deal with finding the value of the variable. The algebra 1 operations with examples are illustrated in the following sections.
Algebra 2 Online Textbook And Generatorinatorics, and number theory, algebra is one of the main branches of pure mathematics
Algebra 2 Representation With Special FactorsBy:-Matthew David
A representation of algebra is a component meant to algebra. Now associative algebra is a sphere, If the algebra is not unital, then it might be prepared thus within a normal method here is no important differentiation among component for the resultant unital sphere, in that the individuality perform through the self plan, with illustration of the algebra.
Algebra PicturesBy:-Matthew David
Algebra is a branch of mathematics, which helps us to find an unknown quantity through mathematical operations. Algebra deals with unknown quantities, which can vary, that are called variables and fixed quantities that are constants. In Algebra, we use mathematical operators to group unknown quantities and form relations with unknown quantities, which are called algebraic expressions and equation. One of the grouping operations is raising the power of the variable or exponentiation.
we know 53=125. This means that logarithm of 125 to the base 5 is 3 and this is written as log5125 = 3
Algebra 1 Test Practice With ExercisesBy:-Matthew David
Algebra is defined as the part of mathematics which includes the study of laws of the operations that includes the equations and various structures including polynomials. In this article we are going to see some practice problems related to the algebra 1 test. Let us see some example problems for easy understanding of algebra 1 test practice.
Algebra The FunctionBy:-Matthew David
In algebra the numbers consider as constants, algebraic expression may include real number, complex number, and polynomials. In algebra several identities to find the x values by using this we can easily find the algebraic expression of the particular function. The sample algebra functions may include in the function of p(x), q(x),… to find the x value of the algebra the functions.
Fun Facts And Fun Games About AlgebraBy:-Matthew David
In math, algebra is a one of the most important part of the mathematics learning of the rules of the operations and relations facts. It includes about the polynomials, equations and algebraic structures. The algebra is combinations of about the analysis and number theory. Generally algebra is a one of the best way for learning about the elementary algebra facts. The elementary algebra is a basic form of the algebra.
Steps To Learn Basic AlgebraBy:-Matthew David
In universal algebra and mathematical logic, term algebra is a freely generated algebraic structure. For case, a signature consisting of a single binary relation, term algebra over a set X of variables is exactly the free magma generated by X. Term algebras play a role in semantics of abstract data types, where an abstract data type declaration provides the signature of a multi-sorted algebraic structure and the term algebra is concrete model of the abstract declaration.
Online Pre Algebra TextbooksBy:-Matthew David
Algebra is a branch of mathematics. Algebra plays an important role in our day to day life. Online pre algebra textbooks cover the four basic operations in algebra such as addition, subtraction, multiplication and division. The most important terms of algebra, variables, constant, coefficients, exponents, terms and expressions are explained in online pre algebra textbooks. We will know the symbols and alphabets in the place unknown values by online pre algebra textbooks. Hence, students can get online pre algebra textbooks.
Sample Pre Algebra Problems To DoBy:-Matthew David
Sample Pre-algebra is a common name for a course in middle school mathematics samples. Pre algebra problems deals with the Factorization of natural numbers, Properties of operations (associatively, distributives and so on), Simple (integer) roots and powers,Pre-algebra often the includes some basic subjects from geometry, mostly the kinds that further understanding of algebra and show how it is used, such as area, volume, and perimeter.
SI Unit Length And MeasurementBy:-Matthew David
Length implies the measure of how long an object or an path is thus we must possess some unit to measure such a important and major fact of our life because one cannot say that the object is 15 long or the path is 15 long because it does means that the object and the path have the same length but it might not true. Thus to measure length we have a specific unit which is called as the SI unit of length and it nothing but meter (m).
Resistance And Measurement SystemBy:-Matthew David
When we apply a potential difference between two ends of a conductor, an electric field will be set up inside the conductor and as a result current flows through the conductor.
The applied potential difference is proportional to the current flowing in the conductor. If V is the potential difference applied and I is the current
Scientific Method Steps And Acid Titration MethodBy:-Matthew David
A scientific method is comprised of asking and satisfying that question. This is made possible making by observations and performing experiments. The experiments executed experiment must be a "fair" test. A fair test refers to only if there is change in one variable, keeping all the other as constant.
Math Help GamesBy:-Matthew David
Generally in math help games, includes some of the play games like finding colors, missing the numbers. The math help games is used to calculate the numbers and finding some of the shapes like how many boxes, how many smiley's, how many balls and how many triangles, etc. Now we will play some of the games in math.
Word Problems In 2nd Grade MathBy:-Matthew David
Word Problems are the main chapter in 2nd grade math. These 2nd grades math involves basic arithmetic word problems. The basic arithmetic operations are add, subtract, multiply and divide. In this topic we have to seen about the addition and subtraction and multiplication and division word problems. Usually word problems are difficult here these 2nd grade math problems are very simple to study.
Math Square Root 12By:-Matthew David
In mathematics, we use a math square root symbol, which is known as radical. The design of radical symbol is (v). Rubicund is referred to a number which is present inside the root (i.e., here x is referred as rubicund). Square root design is deals with the different ways of expressing square root in various designs. Let us see what are the designs are there in square root in brief.
Solving Linear Quadratic SystemsBy:-Matthew David
Linear equation is an equation where the equation contains either constants or group of constant and a single variable. Linear equation has one or more variables.The degree of linear equation is 1.
Quadratic equation is also an equation where the highest degree is two. Quadratic equation general form is Ax2 + Bx + C = 0 where x = variable and a,b and c are constants.
Applying Word Problems In Linear FunctionsBy:-Matthew David
Let us see some of the word problems involving linear functions. In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; in analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. This function is known as "linear" because they are involving the functions whose graph in the Cartesian co-ordinate plane is a straight line.
One Step Linear EquationsBy:-Matthew David
In this page we are going to discuss about one step linear equations . A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables.(Source: From Wikipedia).
Algebra 1 And 2 Practice Problemsinatory, and number theory, algebra is one of the main branches of pure mathematics.
How To Perform AlgebraBy:-Matthew David
Algebra is the one of the important branches of mathematics that deals with the study of the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. In algebra we use letters or symbols to represent a number.Some of the classifications are Elementary algebra, Abstract algebra, Linear algebra, Universal algebra, Algebraic geometry, Algebraic combinatorics.
Algebra 1 Factoring SolverBy:-Matthew David
Algebra is one of the oldest and important topics of pure mathematics. Algebra was first stated by the Greek mathematicians. Algebra consists of constants and variables. Algebra is generally classified into algebra 1, algebra 2 and college algebra. Algebra includes linear algebra, non linear, polynomials, equations, radical expressions etc. When the concept of algebra is understood well then it is easy to solve the algebra problems. |
linear algebra.
...continue to expand their problem-solving skills, in particular, visualization and abstraction.
...gain knowledge and skills to formalize their ideas and express them with a full mathematical rigour.
Content:
Systems of Linear Equations.
Matrices.
Determinants
Vector Spaces
Inner Product Spaces.
Linear Transformations.
Eigenvalues and Eigenvectors.
Course Philosophy and Procedure
Just keep this simple principle in mind: If you are not enjoying this course, if the work is not fun, then something most be wrong. Talk to me right away! This course involves a lot of concepts that easily translate into fairly straightforward (but sometimes lengthy) calculations. Geometry, i.e., visualization is essential. There are also some topics that involve a greater level of abstraction yet there will be plenty of exercises available to check and enhance your understanding of those concepts. You will find that the concepts learned in this course can be applied to many problems in Mathematics and Science.
You should plan to reserve a significant amount of time to study for this course. The material is easy, but the nature of the exercises is such that they are going to be time consuming. Being focused is of utmost importance. Don't rush in doing the problems!
Grading will consist of two semester (100 points each) and the final exam) worth 200 points each. The homework and chapter projects will total to 200 points.
My grading scale is
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.
Final Exam: 12/12 from 12:50-2:50 |
Our users:
The Algebrator software helped me very much. I thought the step by step solving of equations was the most helpful. It was easy to use and easy to understand. I would definitely recommend this to anyone. Thanks, Annie Hines Mary Jones, NY Richard Penn, DE.
I like algebra but was not able to finish the homework on time. Things have changed now for me. Katherine Tsaioun, MA
I am a 9th grade Math Teacher. I use the Algebrator application in my class room, to assist in the learning process. My students have found the easy step by step instructions, and the explanations on how the formula works to be a great help. R.B., New Mexico11-23:
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Why is it important to understand the rules for multiplying and dividing terms with exponents when multiplying rational expressions? |
So you want to include some history of mathematics in your upper-level courses, but you just can't imagine how you can possibly fit anything else in this semester. How will you get to all the topics you want to cover, and still have time for some history?
Instead of giving a lecture on a history topic, or on the name behind a famous theorem, why not let students find the information themselves? Using a discovery worksheet is fun, saves class time, and encourages students to learn things on their own. Some of the answers may be in their own textbooks, or in library books on the history of mathematics, but the activities in this article are designed so that students are encouraged to search the Internet for information. In the process, students will probably be surprised to learn how much material is 'out there' about mathematics and its history and will begin to learn how to separate the online wheat from the chaff. The examples that follow are intended for students in linear algebra and differential equations courses, as indicated, but you can obviously use this idea in any class. This assignment is very flexible: you may assign the worksheets for homework or use them as a class activity; you may have students work independently or in groups.
Students enrolled in a linear algebra class may never have stopped to think that Gaussian elimination was named for someone named Gauss, or that there was a Cramer behind Cramer's Rule. They may be surprised to learn that people have been solving systems of linear equations for thousands of years |
Math Studio description
An easy-to-use math program for high school teachers and students
Math Studio offers you an easy-to-use math application which enables you to easily create 2D and 3D function graphs and animations.
Created for high school teachers and students, Math Studio can be used to help study function, sequence of number, inequality, analytic geometry, solid geometry etc.
Here are some key features of "Math Studio":
· Creating 2D function graphs and animations
· Exponential function, power function, logrithmetic function, trigonometric function, polynomial function, rational function, complex function,etc..
· Functions can be in the form of explicit, parametric, piecewise, implicit and inequality
· Cartesian and polar coordinate systems
· Function graph animation
· Graph of inverse function
· Analyzing functions
· Parity of function
· Maximum, minimum and inflexion
· Solving equations
· Solving inequalities - one-variable linear inequalities, one-variable quadratic inequalities, inequalities with absolute value, two-variable inequalities
· Point of intersection (Solving equation group)
· Area of intersectant field
· Limit
· Graph of derivative function
· Integral
· The length of curve
· Tangent and normal
· Curvature circle
· Creating 2.5D function graphs and animations
· Functions can be in the form of explicit and parametric
· Cartesian and polar coordinate systems
· Function graph animation
· Analytic geometry
· Vector operation - addition, subtraction, multiplication by a constant, inner product
· Line equation - general form, point slope form, gradient intercept form, two-point form
· Perpendicular line or parallel line pass through a point outside of the given line
· Relationship of two lines
· Circle equation - standard form, general form, parametric form
· To find tangent line equation if a point on the circle is given
· To find tangent line equations if the gradient of the line is given
· To find circle equation if center and radius are given
· To find circle equation if center and a point on the circle are given
· To find circle equation if center and the equation of it's tangent line are given
· To find circle equation if three points on the circle are given
· Ellipse and it's geometry characters
· Hyperbola and it's geometry characters
· Parabola and it's geometry characters
· Sequence of number
· Arithmetic progression, sum of the first n items
· Geometric progression, sum of the first n items
· Solid geometry
· Spatial line
· Simple solid, such as prism, pyramid, cylinder, cone, dado and sphere
· Ability to set and modify the properties of function graphs and animations
· Ability to move, zoom in, zoom out and rotate the graphs in plot area
· High quality graph effect, the curve created is very smooth
· Ability to save graphs as msd file or bmp file
· Ability to save animation as AVI file
· Ability to add grid, background and shadow for 2D axis and use pi to label x axis
· Free Tools
· Expression Calculator |
Haymarket Algebra getting to know your specific need and to help you achieve your objectives. Respectfully Yours,
John B., BSME, FE(EIT), MBA, PMP, ITILv3, CSSGBEngineers & Scientists use Mathematics to communicate as much as divers use air to breath. As an Engineer myself, I've come to realiz...Most of my family is deaf, including both parents and both step parents. I also have a sister who is deaf along with her husband and her two kids. My other sister who is hearing, is a sign language interpreter.
...Hence, much will depend on student?s standing. The following is a snapshot of what will be covered in the course: algebraic expressions, setting up equations by translating word problems; evaluating expressions by adding and subtracting polynomials; factoring polynomials (trinomials) using FOIL ... |
Algebra: Equations and Inequalities
- Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines
- Solve multi-step inequalities and graph the solution set on a number line
- Solve linear multi-step inequalities
Algebra: Patterns, Relations and Functions
- Understand that numerical information can be represented in multiple ways:
- Find a set of ordered pairs to satisfy a given linear numerical pattern then plot and draw a line
- Determine if a relation is a function
- Interpret multiple representations using equation, table of values and graph
Geometry: Constructions
- Construct the following using a straight edge and compass: Segment congruent to a segment; angle congruent to an angle; perpendicular bisector; and angle bisector
Geometry: Geometric Relationships
- Identify pairs of vertical angles as congruent
- Identify pairs of supplementary and complementary angles
- Calculate the missing angle in a supplementary or complementary pair
- Determine angle pair relationship when given two parallel lines cut by a transversal
- Calculate the missing angle measurements when given two parallel lines cut by a transversal
- Calculate the missing angle measurements when given two intersecting lines and an angle
Geometry: Transformational Geometry
- Describe, identify, and draw transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations.)
- Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation
Geometry: Coordinate Geometry
- Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change
- Determine the y-intercept of a line from a graph
- Graph a line using a table of values, and by the slope-intercept method
- Determine the equation of a line given the slope and the y-intercept
- Solve systems of equations graphically
- Graph the solution set of an inequality on a number line
- Distinguish between linear and nonlinear equations ax²+bx+c; a=1 (only graphically)
- Recognize the characteristics of quadratics in tables, graphs, equations, and situations |
The Secondary Mathematics program consists of courses that are in line with the Maryland Voluntary State Curriculum, Maryland State Math Core Learning Goals and Bridge Goals the National Council of Teachers of Mathematics Scope and Sequence. After successful completion of the mathematics program students should be prepared for college entrance examinations as well as being able to apply mathematics in real-world situations.
Last modified: 9/12/2007 11:21:37 AM
Accommodations for persons with disabilities attending events/meetings/hearings are available upon request. Requests must be provided as soon as possible. For requests click |
MAA Review
[Reviewed by Mark Hunacek, on 09/06/2012]
A student learning special relativity for the first time must contend not only with counter-intuitive concepts such as time dilation and length contraction but also with fairly cumbersome equations such as the Lorentz equations
x' = γ (x – vt)
t' = γ (t – (v/c2)x)
(where γ = 1/√(1-v2/c2) and c is the speed of light), which relate the position and time of an observer O at rest and a moving observer O'. The novel idea of this slim, succinct book is to seek to replace reliance on these equations with geometric reasoning. Of course the "geometry" here is not the ordinary Euclidean geometry we all learned in high school, and which is based on circles in the sense that the set of all points at distance 1 from the origin is a circle. Here, instead, the geometry is based on hyperbolas: the squared distance of a point (x, y) from the origin is defined to be x2 – y2, so the "unit circle" in this geometry is really a hyperbola. The ordinary trigonometric functions are then replaced by the hyperbolic functions sinh, cosh, and tanh, applied to angles that show up in spacetime diagrams. (Such a diagram plots the position x and time t of an object, with t running up the vertical axis and x moving along the horizontal one; thus, for example, a vertical line depicts an object that stays motionless as time passes.)
The book starts with two introductory chapters, the first of which (very) rapidly contrasts Newton's and Einstein's physics and the second of which functions as a quick overview of the physics of special relativity, including the derivation of the Lorentz equations from the two basic postulates of special relativity (that the laws of physics apply in all inertial reference frames and that the speed of light is the same for all inertial observers). This is followed by two chapters on Euclidean "circle geometry" and the non-Euclidean "hyperbola geometry" that will be used throughout the book, introducing the hyperbolic trigonometric functions from several different (but equivalent) points of view.
The geometry of special relativity begins in earnest with the next chapter, which introduces spacetime diagrams and a particular angle in them called the rapidity, which turns out to be the angle between a worldline and the vertical (ct) axis (ct, rather than t, because we want to measure time and space in the same units). The ideas developed here are exploited in two subsequent chapters ("Applications" and "Paradoxes") which discuss the geometric ideas behind such topics as length contraction, time dilation, the twins paradox, and others.
There are then three (non-consecutive) chapters that struck me as the most demanding in the book: chapter 9, on relativistic mechanics, addresses the relationship between mass and energy and gives insight into the famous equation E = mc2; chapter 11, on relativistic electromagnetism (which, thanks to my dismal background in physics, I only dimly understood) unifies electricity and magnetism; chapter 13 provides a warp-speed look at general relativity.
The book concludes with two chapters that are really just straight mathematics, no physics: one chapter ("Hyperbolic Geometry") discusses various models for hyperbolic geometry, and the other ("Calculus") offers a geometric derivation of the trigonometric and exponential functions.
There are thirteen substantial worked-out examples in the book, all of which struck me as both interesting and illuminating; instead of scattering them throughout the text the author puts them all in three chapters (7, 10 and 12) devoted entirely to them. There are no exercises, but the author does occasionally explicitly leave something to the reader to ponder, such as the resolution of the "manhole cover" paradox in chapter 8.
Despite the fact that there are at least two other books (one quite recent) with titles that are very similar to this book's, Dray's approach really does appear to be novel. Dragon's new book The Geometry of Special Relativity: A Concise Course, for example, seems (based on an admittedly cursory glance) to be addressed to a considerably more sophisticated audience than is this text, and while geometric ideas are certainly mentioned the ones that are discussed in Dray do not appear to take center stage in Dragon. There is also a book by Callahan with the similar-sounding title The Geometry of Spacetime, but that book is considerably different than this one; unlike this text, it covers both general and special relativity (so "geometry" as used in the title of that book really refers to differential geometry, which finds extensive use) and the discussion of special relativity in the first half of the book emphasizes linear algebra notation; matrix computations appear throughout. This is consistent with a primary idea in Dray's book (that the Lorentz transformation is really just a hyperbolic rotation) but the manner of presentation is different. (Anybody planning to look at both books should also note that Callahan, in contravention of what I believe is now standard practice, uses the horizontal, rather than vertical, axis for time.) Finally, I should perhaps mention Taylor and Wheeler's Spacetime Physics, endorsed by the author but unseen by me, which apparently contained a lot of this material in its first edition but left much out of it out of the second.
The question then arises, of course, whether "novel" necessarily means "better". I found that the geometric discussions did shed some light on the underlying ideas, but I also thought at times that the calculations used in them were just disguised versions of the calculations used from a more traditional approach (which, I think, any serious student of special relativity should be familiar with, since these ideas appear so frequently in the literature). So, while I enjoyed reading this book and certainly learned from it, I tend to think that it would serve best as a supplemental text for a course in special relativity rather than as a main text. (The very succinct writing style, and the total lack of homework exercises, also influenced this opinion.) The author himself may think this, because he writes in the preface that the book "is not intended as a replacement for any of the excellent textbooks on special relativity" but is intended as an introduction "to a particularly beautiful way of looking at special relativity… encouraging students to see beyond the formulas to the deeper structure." This goal, I think, has been met: on more than one occasion as I read this book I found myself looking at other texts to compare discussions, and I generally found that the process seemed to have a synergistic effect: I got more out of both books by doing this. This is unquestionably a book that anybody who teaches special relativity will want to look at. |
Freeware
| September 11, 2002
Challenges for Teaching Electricity and Electronics, Version 4.0g
Source: ETCAI Products
Challenges for Electronics is a suite of seven educational programs for electricity, electronics and math. The titles of the seven programs are Basic Circuits Challenge, DC Circuits Challenge, AC Circuits Challenge, Trigonometry Challenge, Digital Challenge, Solid State Challenge and Power Supply Challenge. Each program contains several interactive activities. Activities for both circuit analysis and troubleshooting are included. The programs grade and correct all student work immediately. Student scores can be stored on diskettes or printed. The material is suitable for use as a supplement to classroom or tutorial instruction. The programs can also be used as a refresher course for employees who have already had basic electrical or electronics training. The programs run for ten executions of any time length. After ten executions, the programs become limited capability demos |
Find a Lucerne, CO took a course in Differential Equations at St. Petersburg University and passed this course with "B". I defended my PhD on mathematical modeling based on the use of Partial and Ordinary differential equations in Earth scienceIt's a necessary prerequisite for all future math and science classes, and nearly every high school and college requires it as a core course. Fortunately, because it is a class that's mandatory, it's also class that everyone should be able to pass. Sometimes it just takes a little bit of help. |
Useful computer tips and software giveaways .
Math is a hard task for most of the school and college students. If you are struggling to complete the mathematics school work or assignments, then Microsoft Mathematics can be of great help.
Microsoft Mathematics 4.0 is an free award-winning education tool which focuses in solving mathematical problems ranging from basic mathematics to complicated precalculus. It consists of various powerful mathematical modules to offer help on pre-algebra, algebra, trigonometry, physics, chemistry, and calculus. One of my favorite features is step-by-step solutions to assist students in learning the fundamentals of resolving mathematical problems. They show you the individual steps to a solution, with basic explanations to help you understand how to solve the problem. It's great when you get stuck and you need someone to explain it to you.
The previous versions of Microsoft Mathematics which was known as Microsoft Math was a standalone purchasable product which require product activation (sold for around $20). Now the new Version 4 is available free to download from Microsoft Download Center, or from the direct download link below:
You may also like to install additional add-in to plot graphs in 2D and 3D, calculate numerical results, solve equations or inequalities, and simplify algebraic expressions in your Word documents and OneNote notebooks.
Download the free Microsoft Mathematics Add-in for Word and OneNote From here.
2 Comments for this entry
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Math for the Biology Major or Pre-Med Student
A biologist or medical professional must be able to think mathematically (analyze graphs, interpret quantitative information, use clear logical patterns). An early decision to get a strong mathematical background, particularly in the core areas of calculus, probability and statistics, linear algebra (vectors, matrices, systems of equations), and computer programming, will multiply a student's career options.
The Formal Requirement for the Biology Major
Biology majors are required to take one semester of calculus plus a second math or statistics course. Sample programs include:
two semesters of calculus, such as MATH 1110-1120 or MATH 1110-1220;
one semester of calculus plus a course in finite mathematics, such as MATH 1105-1106;
Medical School Entrance Requirements
College work in mathematics is required by some medical schools and recommended by almost all. A very few medical schools require one year of calculus. Also, a very few require one semester of statistics. See the book Medical School Admissions Requirements or individual medical schools' web pages to verify premedical requirements.
Calculus and Further Study
Students who may take more than one year of mathematics should definitely start with two semesters of calculus. (See First-Year Calculus.) The following options provide a good introduction to the core mathematical areas most useful in the biological sciences. Courses under Option 2 are a bit more challenging than those under Option 1.
Core Mathematical Areas Most Useful in the Biological Sciences
Subject
Option 1
Option 2
Calculus
MATH 1110-1120
MATH 1110-1220
Multivariable calculus
MATH 2130
MATH 2220
Linear algebra
MATH 2310
MATH 2210
Probability and statistics
MATH 1710
MATH 4710-4720
Computer programming:
CS 1110, 1112, 1114, or 2110
Of course, much more is possible and in some cases necessary. Further study in mathematics could lead to a math minor, which is available to students in all colleges. MATH 4710-4720 combined with two other upper-level math courses, one in algebra and one in analysis, would result in a math minor, for example. |
Master Math:: AP ready to master the concepts of AP Statistics and ace your exam! Master Math: AP? Statistics is a comprehensive reference guide written specifically for AP Statistics students, covering all the topics of AP Statistics in a simple, easy-to-follow style and format. Suitable for a wide variety of ability levels, this book explains and clarifies the various concepts of AP Statistics including exploring data, sampling and experimentation, anticipating patterns, and statistical inference. The example problems in each chapter are written with the AP Statistics Exam in mind to help you understand the concepts and learn how to effectively answer the exam questions. You'll also find useful appendices that help with exam preparation, including all the tables and formulas that are given and needed, as well as a quick-reference summary of assumptions and conditions for inference. A helpful glossary will help you brush up on terminology. Master Math: AP? Statistics is an invaluable resource for anyone studying and preparing for the AP Statistics Exam. |
MAT
140
- Precalculus
This course emphasizes the algebra and concepts of functions. Students will learn the properties and graphing techniques for different types of functions including: linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. Students will also learn to solve a variety of real world problems that rely on a number of different problem solving strategies and an understanding of these different types of functions. |
Publisher review: GeoGebra is a very useful mathematics tool for education in secondary schools, which brings together geometry, algebra and calculus.
GeoGebra is also a dynamic geometry system, meaning you can do constructions with points, segments, vectors, lines, conic sections as well as functions and change them dynamically afterwards.
On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.
These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa. GeoGebra key features:
Captions enabled for all objects
New option "Force Reflex Angle" forces angles to be between 180 and 360 degrees
Pressing toggles the focus between the Input Bar and the Graphics View
Comparing objects of different types doesn\'t return an error, can now compare Text and Image objects
If the Points Export_1 and Export_2 exist, they will be used to define the export rectangle (Export_1 and Export_2 must be within the visible area)
Checkbox now consistent across all platforms
Options -> Checkbox Size -> Regular/Large
Perpendicular check added to "Relation between two objects" Tool
Messages from "Relation between two objects" Tool rewritten
Angular Bisector Command and Tool renamed to Angle Bisector
Line Bisector Command and Tool renamed to Perpendicular Bisector
BMP import
Unicode fonts used in LaTeX equations
LaTeX equations exported at full resolution
in SVG and PDF export, option to export text as editable text or shapes. Stores the text either as text (lets you edit the text in eg InkScape) or as bezier curves (ie guaranteed to look the same even if the correct font is not installed). |
Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers, integers, rational numbers, and irrational numbers)
- Write numbers in scientific notation and translate back into standard form
- Find the common factors and greatest common factor of two or more numbers
- Determine multiples and least common multiple of two or more numbers
- Determine the prime factorization of a given number and write in exponential form
- Simplify expressions using order of operations - Note: Expressions may include absolute value and/or integral exponents greater than 0.
- Add, subtract, multiply and divide integers
- Recognize and state the value of the square root of a perfect square (up to 225)
- Determine the square root of non-perfect squares using a calculator
- Identify the two consecutive whole numbers between which the square root of a non-perfect square whole number less than 225 lies
Algebra:
- Translate two-step verbal expressions into algebraic expressions
- Add and subtract monomials with exponents of one
- Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation
- Solve one-step inequalities (positive coefficients only)
Geometry:
- Calculate the radius or diameter, given the circumference or area of a circle
- Calculate the volume of prisms and cylinders, using a given formula and a calculator
- Identify the two-dimensional shapes that make up the faces and bases of three-dimensional shapes (prisms, cylinders, cones and pyramids)
- Determine the surface area of prisms and cylinders, using a calculator and a variety of methods
- Find a missing angle when given angles of a quadrilateral
- Identify the right angle, hypotenuse, and legs of a right triangle
- Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle
- Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator
- Graph the solution set of an inequality (positive coefficients only) on a number line
Measurement:
- Calculate distance using a map scale
- Convert capacities and volumes within a given system
- Calculate unit price using proportions
- Draw central angles in a given circle using a protractor and display data (circle graphs) |
This book is the ultimate math resource or home which has everything you need for math success. Also includes a handy Almanac with math prefixes and suffixes, problem-solving strategies, study tips, guidelines for using spreadsheets and databases, test-taking strategies, helpful lists and tables, and more. |
Thursday, November 24, 2011
Tuesday, November 22, 2011
Who uses it? Isaac uses it. Actually, all of my kids switch from Horizons Math to Saxon 54when they hit the Saxon 54 book. As far as I can tell, Emma will be transitioning into Saxon as planned next year.
What do you get? When I purchased the Homeschool Packet, I received the student text, solutions manual, answer key, test booklet. I also purchased the DIVE CD and Lesson Plan from MFW. If you plan to purchase the DIVE CD, be sure you check the editions of both the CD and the text to make sure they match.
Where can you get it? Saxon math is published by Saxon Publishers. You can purchase the curriculum from Saxon, online retailers such as Amazon or CBD, and online curriculum stores like My Father's World and Sonlight.
How much is it?
Prices below are for Saxon Algebra 1/2. Prices vary for each text.
Homeschool Kit – $74.70 (student text, tests, answer key)
Solutions Manual – $40.20 (optional)
Lesson Plan Booklet – $15.00 (Sold exclusively by My Father's World. According to their site: These plans assign the problems in each lesson that are most important to complete.)
Why? It works for us! It does not work for everyone. My kids do okay with a black and white, no-frills text. They need the spiral teaching style of Saxon. This helps them keep concepts fresh in their minds.
How we use it: Isaac starts out each day correcting his work from the day before if he scored less than 85%. I do this for a few reasons. Firstly, I want him to correct his mistakes so that I can see that he has grasped the concept before moving ahead. It keeps him from rushing through his lessons if he knows he'll have to correct his mistakes the next day. By not making him correct his lesson if he scores at least an 85%, it keeps him from feeling 'stupid' (his word not mine) if he makes an occasional mathematical error – missing a multiplication fact or adding wrong.
After I check his corrections to make sure he got them right the second time, he does his DIVE CD lesson. These CD's present each day's lesson on a digital white board. Isaac can hear the instructor 'teach' each lesson while watching him work the sample problems on the white board. After the problem is written on the board, Isaac pauses the CD and works the problem in his notebook. Then, he hits play to see the answer. If his answer doesn't match the CD answer, he is to go back and listen to the 'lecture' part over again. If he still doesn't understand, he brings it to me.
Last year, Isaac got about 1/3 of the way through Saxon Algebra 1/2 without the DIVE CD. He hit a wall and got to the point where he didn't want to have me lecture/teach him the math lesson. Isaac thought I was being critical of him when I was correcting him. It ended with LOTS of tears from both of us. It wasn't productive for either of us. So, we put the book away and didn't do math for the rest of the year. So far, this year has been a complete turn around. He has gained confidence in his math abilities. He enjoys the CD, because someone else is teaching him.
I use the lesson plan booklet from My Father's World. I am not comfortable knowing which Saxon problems can be skipped (and if you do any research online you will see that parents are STRONGLY discouraged from skipping problems). However, I used the lesson plan book a few years ago with Logan, and it worked out just fine. One of the things Isaac hated about Saxon last year was that he had to do 30 problems each day. Now, the book tells him which problems to do. Some days he has 10-15, and other days he has 20-30. For some reason, if the book is telling him to 30 problems there is much less complaining! That is worth $15 in my opinion!!
After the DIVE CD lesson, Isaac does the practice problems in his text and I check them. Then, he's off to complete that day's lesson.
My recommendation: If you're looking for a no-frills spiral approach to math, this may be what you're looking for.
Here's the order we (plan to) do Saxon Math:
4th grade – Saxon 54
5th grade – Saxon 65
6th grade – Saxon 76
7th grade – Saxon Algebra 1/2 (Isaac is a year behind this schedule due to the issues I mentioned above.)
8th grade – Saxon Algebra 1
9th grade – Saxon Algebra 2
10th grade – Saxon Advanced Math
11th grade – Saxon Calculus (over 1.5 – 2 years if necessary)
12th grade – College Algebra at the community college (dual enrollment) upon the completion of Calculus
*This isn't a Crew review or a solicited review of any sort. It's my honest opinion of our math curriculum - that I purchased -, and I wanted to share it with you.
Monday, November 21, 2011
We are not doing school this week. I am babysitting Monday – Wednesday. I'm cleaning house and running errands to prep for Thanksgiving at my house.
My in-laws are driving down from Missouri on Wednesday – all 6 of them. A close friend and her family are coming over for Thanksgiving dinner – all 5 of them. My sister will also be coming over for dinner. We do the traditional Thanksgiving fare – turkey, ham, rolls, stuffing, cranberries, veggies, mac-n-cheese, and TONS of desserts.
Friday, I'm hoping to get a little shopping done.
Saturday, we are meeting friends in our downtown area to attend our local Dickens of a Christmas festivities. I can't wait!
Sunday, I'll rest up for the next week of school. Maybe, we'll get the Christmas tree put up, too.
Sunday, November 20, 2011
Well, this turned out to be a pretty productive week! And, it's a good thing since we are off next week for Thanksgiving. I'm babysitting one of my afterschool girls Monday – Wednesday of next week so we won't be doing any schoolwork.
learned about verbals – infinitives and gerunds – and learned how to diagram them
created some Monet-inspired artwork
listened to Vivaldi's Ring of Mystery
Logan:
finished reading Around the World in Eighty Days and wrote an approach paper
learned about Russia and took a European countries and capitals map quiz
created some Matisse-inspired artwork
listened to Vivaldi's Ring of Mystery
I jotted down some notes as I was teaching this week. Here are a few things I observed:
Isaac needs to memorize the prepositions so he can recognize prepositional phrases and objects of the preposition easier.
Emma needs to memorize her times tables to eliminate the need to count on her fingers.
Isaac and Emma have asked to do Latin and Spanish together. So, we will be stopping where we are so Isaac can catch up in Latin and Emma in Spanish. We will do them together for the remainder of the year.
Logan needs to really work on managing his time wisely. He got pretty behind this week on his homeschool and college class work, and pulled an all-nighter to catch up on Thursday. I don't want this to become a habit so we will be working on time management after Fall Break.
Friday, November 18, 2011
After reading Alice's Adventures in Wonderland, Isaac read in his Lightning Literature text that Lewis Carroll would often sit in a room with his eyes closed and listen to the sounds around him. He would then turn those thoughts and sounds into what-if questions. Many of those questions wound up becoming smaller stories in Alice's Adventures in Wonderland.
For Isaac's writing assignment, he chose to write 50 what-if questions in the style of Lewis Carroll. I had to share some of them here, because they're funny! My brain just doesn't go to those kinds of places. Isaac's questions had me laughing out loud. I could never in a million years have come up with questions like these.
What if the whole world was Disney World?
What if Tom ate Jerry?
What if all the parents burnt down Chuck E. Cheese?
What if the presidents on money talked?
What if calculators told you to figure it out?
What if snakes turned into jump ropes?
What if the Beatles lived in a red submarine?
What if eggs and ham were really green?
What if blondes made fun of brunettes?
What if someone did not laugh at all while reading this?
There really is never a dull moment in this house!! And, I like it that way!
Wednesday, November 16, 2011
Tuesday, November 15, 2011
The College Prep Genius curriculum helps prepare students to master the SAT. The $99 set (a 25% discount) comes with the Mastering the SAT Class DVD, the College Prep Genius Textbook and a Student workbook.
My oldest, a senior in high school, has taken the SAT twice and has plans to take it one more time in December. I have him working through this program in preparation for his upcoming test.
Logan and I are both very impressed with this program. Jean Burk takes students through learning the secrets of the SAT. Logan had no trouble at all learning the acronyms that are used to decode the SAT.
The twelve chapters cover all the sections of the SAT. According to the website, the program teaches students to write a great essay in 15 minutes and how to eliminate 2-3 answers immediately.
Logan is really excited to see how high he can raise his score after completing this program. He's hoping to qualify for a scholarship based upon his SAT scores. He's prepping with College Prep Genius while I'm praying and keeping my fingers crossed!
Logan went on his first college tour Friday. Matt took the day off work, and they visited Dallas Baptist University. This is Logan's first-choice school. He has quite a few friends who attend DBU, and is really hoping to secure enough scholarship money to make it happen.
Monday, November 14, 2011
This weekend we had the opportunity to watch a World War II reenactment. I must say that I wasn't very excited about the field trip initially. I was, however, excited to meet up with some friends I hadn't seen since September. It turned out, however, to be pretty interesting. The kids enjoyed themselves, too. Here are a few pics from our trip. |
Introduction This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound PropertyIntroduction |
Algebra And Trigonometry - 01 edition
ISBN13:978-0534434120 ISBN10: 0534434126 This edition has also been released as: ISBN13: 978-0534380298 ISBN10: 0534380298
Summary: James Stewart, the author of the worldwide best-selling calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this book to address a problem they frequently saw in their calculus courses. Many students were not prepared to "think mathematically" but attempted to memorize facts and mimic examples. Algebra and Trigonometry was designed specifically to help readers learn to think mathematically an...show mored to develop true problem-solving skills. Patient, clear, and accurate, the text consistently illustrates how useful and applicable mathematics is to real life. The new book follows the successful approach taken in the authors' previous books, College Algebra, Third Edition, and Precalculus, Third Edition. ...show less
0534434126 |
Combinatorial Problems and Exercises
9780821842621
ISBN:
0821842625
Pub Date: 2007 Publisher: American Mathematical Society
Summary: The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems.
Ships From:Boonsboro, MDShipping:Standard, Expedited, Second Day, Next DayComments:Brand new. We distribute directly for the publisher. The main purpose of this book is to provide... [more]Brand new. We distribute directly for the publisher. The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems (apart from some general comments at the beginning of each chapter). In the second part, a hint is given for each exercise, which contains the main idea necessary for the solution, but allo [less] |
Special Technology Requirements
Note:
These are course-specific requirements that go above and beyond the Provider Baseline Technical Requirements.
The school or student is responsible for providing:
This class uses Apex Learning online curriculum. Each student will run a system checkup during the Getting Started activity to insure their computer has all the required features and settings; Refer to the Apex Learning System check-up:
Students will need the following materials to complete the course:
The necessary computer hardware as defined at the APEX website
First Class e-mail accounts for both the parent and student will be provided
Speakers to access the audio of the text
A printer for the worksheets and a notebook
Materials to be ordered via the DLD
The DLD Registrar may order the following materials via the DLD upon registering the student:
No additional materials required for this course.
Description
This credit retrieval course is for students who have already taken this class and not earned credit. A diagnostic assessment, given at the beginning of each unit, indicates what standards have been met, and what skills need to be practiced and demonstrated. Based on the results of the diagnostic assessment, an individual plan for completing the course is developed by the teacher and communicated to the student. Upon successful completion of the course, the student earns a passing grade (C) and .5 credit in Algebra 2-1. This first semester of algebra 2 credit retrieval course covers the Algebra 2 Washington state standards: select and justify functions and equations to model situations, solve problems involving exponential and quadratic functions, explain how number systems are related, and simplify rational expressions. Because high school students have unique needs and experiences, CompassLearning ensures that students know where they are, while challenging them to grow. Odyssey High School Math focuses on foundational skills to support learners, emphasizes repetition and practice of key skills, reinforces study habits, including note-taking, to sharpen students� comprehension, and covers National Mathematics Advisory Panel�s concepts for success in algebra. |
Course Tuition:
$2.00 and 4 1st-class stamps
Also available for free by email or online
Please note: The Cadet Academy Courses are courses specifically
designed for the youth members of Starfleet, or those member due to some
handicap are unable to complete the standard course, if you are adult
without limitations, do not apply for the Cadet Academy courses.
CCOM101 - Cadet Elementary Math
This course is designed for grades 4 to 7. Topics covered include: Place Value, Average of a set of numbers, Roman Numbers, Fractions, Rounding, Division and Multiplication.
CCOM102 - Cadet Secondary Math
This course is designed for grades 8 to 10. Topics include: Metric system conversions, Exponents, Factorials, Permutations, Least Common Multiples, Cube Roots, basic Geometry and Algebra, converting temperatures between Celcius and Fahrenheit. Formula sheet is provided. Using a calculator would be a good idea for this course.
CCOM103 - Roman Numbers
This course is designed for grades 4 and up. Topics covers adding and subtracting with ROman Numbers.
CCOM104 - Cadet Math: Pre-Algebra
This course is designed for grades 6 and up. Topics covered are Factors, Exponents, Ratios, Prime and Composite Numbers, Greatest Common Factors and Least Common Multiples.
SOURCE
MATERIALS NEEDED:
None |
Analysis
Analysis is the study of various concepts that involve the idea of taking limits, such as differentiation, integration, and notions of convergence. Princeton's emphasis on analysis is reflected by the fact that two of the three introductory courses for math majors (MAT215 and MAT218) deal with the subject. Analysis has applications ranging from physics to number theory, and underlies many branches of applied math. After taking the introductory courses, most students interested in analysis proceed to the four core analysis courses called the "Stein sequence", described below. The department also offers courses on the applications of analysis to other fields, including MAT 407 (Mathematical Methods of Physics) and MAT 415 (Analytic Number Theory).
MAT 215: Single Variable Analysis
This introductory class covers the first eight chapters of Walter Rudin's "Principles of Mathematical Analysis". Many of the concepts such as differentiation and integration might already be familiar to students. However, the emphasis of the course is on the rigor of proofs. The second chapter on point-set topology is especially crucial in the sense that it lays foundation for the rest of the course as well as for any future analysis classes. It might be beneficial to go through every definition a few times to fully internalize the ideas. A general difficulty in studying this class is about writing proofs. In the first half of the class, most of the proofs more or less "write themselves" once you have unraveled the definitions, so the most important part is again trying to understand what the statement is really saying. For instance, as soon as you write out definitions of continuity and closedness, it should be quite clear that "the zero set of a continuous real function is closed". Chapter 7 of Rudin's book touches on a subtle but important issue in analysis: when can you interchange two limiting processes? For example, if a sequence of function f_n converges pointwise to a function f, is the limit of the integral (by definition a limit process) of f_n the same as the integral of the limit of f_n? This turned out to be a recurring theme in many later courses, and the study of which leads to many fascinating ideas such as uniform convergence and convergence theorems of Lebesgue integration. MAT 218: Analysis in Several Variables
This course is the last in Princeton's introductory sequence for math majors. It builds on the single variable analysis theory developed in MAT 215, but also uses linear algebra tools from MAT 217, such as vector spaces, linear transformations, and determinants. In recent years, MAT 218 has been the least popular of the three introductory courses, and in fact the math department is considering switching to a two-semester introductory sequence. More information about the content and structure of MAT 218 can be found here on the math department's website.
Analysis Sequence [Show]Analysis Sequence [Hide]
At the core of Princeton's program in analysis is the analysis sequence, known as the "Stein sequence." The Stein sequence is based on a series of four books written by Elias Stein, a famous analyst and professor emeritus at Princeton. The courses in the Stein sequence are Analysis I: Fourier Analysis, Analysis II: Complex Analysis, Analysis III: Real Analysis, and Analysis IV: Special Topics in Analysis. It is convenient, but not necessary, to take these classes in order. For example, both the Complex Analysis and Real Analysis courses deal with aspects of Fourier analysis. Having said this, the courses are advertised as being self-contained, so taking them in other orders is also fairly common. MAT 325: Fourier Analysis and Partial Differential Equations
This course covers the fundamental concepts in Fourier series and Fourier analysis, as well as their applications to differential equations, number theory, physics, and other topics. It begins with a study of the behavior of a vibrating string, which motivates the idea of decomposing a periodic function into an infinite sum of sines and cosines. It then discusses the basics of Fourier series, such as the Dirichlet and Fejér kernels, Parseval's theorem, and the Riemann-Lebesgue lemma, with a particular focus on issues of exactly when and how the Fourier series of a function converges to that function. The course then moves on to Fourier analysis, covering the Fourier transform, the inverse transform, Plancherel's theorem, and the Poisson summation formula. These results are first shown on the real line, and are then easily generalized to n-dimensional space. Finally, the course studies Fourier analysis on an arbitrary abelian group, using the particular case of the unit group of Zn to provide an elegant proof of Dirichlet's theorem on arithmetic progressions. Along the way, these results are applied to various interesting topics, such as the solutions of the wave and heat equations, the isoperimetric inequality, the existence of a function which is continuous everywhere and differentiable nowhere, and the Heisenberg uncertainty principle.
The course closely follows Elias Stein and Rami Shakarchi's textbook Princeton Lectures in Analysis I: Fourier Analysis. The only true prerequisite for the course is a solid understanding of high school calculus, but an introductory analysis course such as MAT 215 is certainly helpful. Although Fourier analysis is more naturally done in the context of Lebesgue integrable functions, Stein and Shakarchi choose not to introduce measure theory until Book III of their series, so this course is taught within the context of Riemann integrable functions. The course grade is based on weekly problem sets, which tend to consist of exercises from the textbook, as well as a midterm and final exam. Professor Stein usually teaches the course every other year, and he is notorious for including final exam questions that require you to write out proofs of major theorems from the textbook, so it is important to set aside some time during reading period to memorize the most important proofs. MAT 335: Complex Analysis
The second course in the Stein sequence deals with complex analysis. This branch of analysis deals with analytical properties of holomorphic functions of complex variables. A function complex-valued function of a complex variable is holomorphic at a point if it has a complex derivative there. This turns out to be a strong condition, one which makes complex analysis a much richer theory than, say, analysis of a single variable. The many beautiful properties of holomorphic functions give complex analysis a unique flavor. MAT 335 follows Stein's well-written Complex Analysis. The course usually covers most of the book's ten chapters, in order. The first three chapters establish the many elegant basic properties enjoyed by a holomorphic function. Chapters 4, 5, 8 are three stand-alone chapters that cover fascinating individual topics in complex analysis. Highlights include Hadamard's factorization theorem (entire functions are more or less determined by their zeros) and Riemann mapping theorem (given any two proper simply connected domains in the complex plane, there is some holomorphic function mapping one bijectively to the other). Chapter 6, 7, 9, 10 are strongly flavored in number theory. It is a wonder that complex analysis has become such a successful tool in tackling many problems in number theory, a field concerned with the properties of integers. Chapters 6 and 7 introduce the zeta function (the link between number theory and complex analysis) and prove the Prime Number Theorem, one of the milestones in analytic number theory. Chapters 9 and 10 briefly touch on the vast subject of elliptic functions and theta functions. MAT 425: Real Analysis
The third course in the analysis sequence is Real Analysis. This course adds another layer of sophistication to the theories of integration and differentiation covered in MAT 215, extending them to a more general context. The course covers the first five chapters of Professor Stein's book Real Analysis: measure theory, integration, and Hilbert spaces.
The first chapter of the book establishes the Lebesgue theory of measure, which underlies the theory of Lebesgue integration. The measure of a set is in some sense its "volume." The Lebesgue measure drastically generalizes this intuition. For example, the measure of the set of rational numbers is zero. The material in this chapter is similar to that of MAT 215, with its technical, but often elegant, epsilon-delta arguments.
The second chapter covers the theory of Lebesgue integration. The starting point of this theory is the characteristic function of a set, which is 1 on the set and 0 elsewhere. The integral of such a function is defined to be the measure of the corresponding set. The Lebesgue integral is then built up from this foundation. The generality of the Lebesgue measure makes the Lebesgue integral a significant improvement on the Riemann integral. For example, the characteristic function of the rationals on [0, 1] is not Riemann integrable, but is Lebesgue integrable. The rest of chapter 2 discusses the properties of the Lebesgue integral.
Chapter 3 explores analogues of the fundamental theorem of calculus in the setting of the Lebesgue integral. An important result is the Lebesgue differentiation theorem, which states (loosely) that the derivative of the integral of a function is the function itself. It turns out that it is more difficult to understand the integral of the derivative of a function. The class of functions of bounded variation is introduced; it is these functions whose derivatives are integrable. A stronger assumption called absolute continuity is needed to guarantee that the integral of the derivative is the function itself.
Chapters 4 and 5 deal with Hilbert spaces, which are vectors spaces endowed with inner products, separability, and completeness. Chapter 4 covers Hilbert spaces and operators on these spaces, two very fundamental topics. Chapter 5 covers applications of Hilbert spaces, including L^2, the space of square-integrable functions, which is the link between Hilbert spaces and integration theory. MAT 425 is an interesting course, but also a very difficult one for many. The course proceeds at a rapid pace, and thus reading before lecture is strongly advised. The problem sets are the hardest part of the course, and doing well on them requires a significant time commitment. The problem sets are very important in understanding the course material, but some problems are quite tricky. Working in groups on the problem sets is advisable, both to split the difficulty and to discuss the material with other people. MAT 520: Special Topics in Analysis
This topics course varies in content from year to year. In Fall 2012, this course will cover functional analysis, following Prof. Stein's recently published book on the subject.
Other Analysis Courses [Show]Other Analysis Courses [Hide] MAT 314: Introduction to Real Analysis
The course goes over the basics of measure theory and Lebesgue integration. It begins with a quick introduction of algebras, sigma algebras, measures and their properties. Then the course develops the basic theory of Lebesgue integrals over the real line including Fatou's Lemma, Dominated Convergence Theorems, Monotone Convergence Theorems, the Egorov Theorem, and others. In the middle of the semester the course discusses abstract measure spaces and integrals with a focus on the Carathéodory extension theorem, Hahn decomposition theorem and Radon-Nikodym theorem. In the second half of the semester the course discusses Lp spaces, inner product spaces with an emphasis on L2. Near the end of the semester the course goes over the basic properties of Hilbert spaces and Fourier series.
The course is intended for sophomores and juniors who require a basic introduction to real analysis. The material is presented carefully and rigorously; students are consistently motivated by examples that test their understanding. For the most part the course follows the book by Royden – "Real Analysis", although the topics are not followed in class in the same order as they appear in the book. The material on Fourier series is usually taken from a different text and Professor Warren often provides additional material either in lecture or in homework sets.
There are weekly problem sets that account for about 30% the grade in the course. Most of them aim to improve student understanding of the theory and can be quite challenging. Lectures are usually spent proving results, emphasizing their role in the theory as well as the techniques employed in the proofs. In addition, there is a midterm and a final (both take-home) that account for 70% of the grade. The exams are usually much more difficult than the homework sets, but the material tested is always within the range of what has been taught.
The course is exceptionally challenging for people who have not taken MAT 215 (or equivalent) before as basic properties of sequences are assumed to be known. Although MAT 202 and MAT 201 (or equivalent) are listed as prerequisites, there is hardly any use of linear algebra or multivariable calculus, although it is still advisable to check this with Professor Warren. MAT 407: Mathematical Methods of Physics
The content of this class varies from year to year depending on the instructor. In particular, it depends on whether a physicist or a mathematician teaches the class. The following are descriptions of MAT 407 for two recent years. Professor: Elliot Lieb
Prof. Lieb covered basic Hilbert Space theory, distributions, Fourier transforms, and briefly introduced the class to unbounded operators, which are important for quantum theory. Then, he spent a significant amount of time studying Trace class operators and their utility in quantum statistical mechanics. Prof. Lieb explored mathematically the concept of entropy in thermodynamics, along with some important inequalities such as the Peierls-Bogoliubov. Towards the end of the course, he also discussed some representation theory and showed where it was relevant for quantum mechanics. The course was fast paced, but provided a great overview of some important working fields in mathematical physics. For additional inspiration, a few of the problems Professor Lieb assigned were related to some of his papers. These papers can be found in the book Inequalities: Selecta of Elliot Lieb. Professor: Chris Herzog
Prof. Chris Herzog, a string theorist who is no longer at Princeton, taught MAT 407 in Spring 2011. The class was structured as a broad survey of mathematical topics applied to problems in physics. Topics included linear algebra and basic operator theory (quantum mechanics), linear ordinary differential equations and Green's functions (related to Prof. Herzog's research in AdS/CFT correspondence), special polynomials (e.g. of the Hermite, Laguerre, Legendre varieties, each with applications to several areas of physics), group theory, basic representation theory – in particular character theory (determining the vibrational modes of molecules from their symmetries), and Lie algebras (elementary particle physics). Coverage emphasized breadth over depth; in the interest of time, proofs of only the most important results were presented. The prescribed textbook was Sadri Hassani's Mathematical Physics, but Stone and Goldbart's Mathematics for Physics is also a very good reference. As for prerequisites, it is sufficient to have knowledge of linear algebra, elementary quantum theory, and complex analysis. MAT 451: Advanced Topics in Analysis
Prof Lieb teaches this class using his textbook "Analysis". The class is quite self-contained. It starts by reviewing concepts of measure theory and Lebesgue integration. Then it goes on to touch a wide variety of topics, including L^p spaces, integration inequalities, Fourier transformation etc. If the first half of the course focuses more on "tools" that analysts should master, the second half shifts more towards using of these tools to tackle problems in partial differential equation and mathematical physics. The highlight of the course is the study of distribution and Sobolev spaces and their application to solving some partial differential equations such as the heat equation and the Schrodinger's equation. MAT 390/MAT 391: Probability Theory and Random Processes
These two probability theory courses are taught over two semesters by Prof Sinai. These two classes count towards the math department's real analysis requirement because they have a substantial measure theory component. Hence, like MAT 425, they also offer an introduction to the theory of measure and Lebesgue integration. Please see the Probability and Statistics section for detailed summaries and discussions of these two courses. MAT 415: Analytic Number Theory
This course is devoted to analytic techniques in number theory. Please see the Number Theory section for a discussion of this course. MAT 427: Ordinary Differential Equations
This differential equations course recently underwent a switch from a 300-level course to a 400-level course, which suggests that it will become more oriented towards theory. However, not much more is known at this time. Please check the math department page and the registrar's page for more information. |
Synopsis mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
Presents major ideas and branches of pure mathematics in a clear, accessible style
Defines and explains important mathematical concepts, methods, theorems, and open problems
Introduces the language of mathematics and the goals of mathematical research
Covers number theory, algebra, analysis, geometry, logic, probability, and more
Traces the history and development of modern mathematics
Profiles more than ninety-five mathematicians who influenced those working today Imre Leader, June Barrow-Green, Timothy Gowers |
All compensation and benefits administration requires the application of basic mathematical skills. This course prepares business professionals for basic quantitative analysis using mathematical calculations for compensation and benefit program design, administration, and evaluation. Here you will review math basics, including operations, measurement types, and percentages. Then you will review graphing population data, a very useful technique in compensation and benefit plan design. Survey samples will be discussed, along with ranges, statistical techniques, hypothesis, and inference. Upon completion of this course, you will be prepared for more advanced Distance Learning Center online quantitative courses, including Course 19: Quantitative Methods Used in Salary Administration |
Algebra 1 Part 1 (2816 A 2817B) Grade 9
1 year
This course covers topics in Algebra I at a slower, more in-depth approach
with emphasis on the review of basic mathematical skills. Students are
required to complete projects.
Algebra 1 (2770 A 2780 B) Grades 9-10
1 year
This course Is designed to provide a firm foundation for future growth in mathematics. Topics include: relations, functions, linear equations annd inequalitites, systems of equations, ratio and proportion, similarity, measurement, coordinate, geometry, as well as an introduction to non-linear functions and polynomials. Students will be required to solve problems graphically, algebraically and symbolically. Use of a TI-83/84 graphing calculator, nightly homework, and semester projects are required.
Algebra 1 (H) (2800 A 2810 B)
Grade 9
1 year Prerequisite:A "B" or better in Pre-Algebra.
This course is designed for those students whose mathematics background indicates the ability to work at an advanced level. Students are expected to handle the Algebra concepts at an increased depthof work and at a more rigorous pace. Students willl be required to solve problems graphically, algebraically and symbolically. Use of a TI-83/84 calculator, nightly homework, and semester projects are required.
Financial Math (3072 A 3073
B)
Grade 12
1 year
This course is designed to provided a comprehensive review of the fundamental
principles of mathematics including percentages, decimals and fractions with emphasis on calculator usage. This course ultilizes the computations of earned pay, banking practices, income tax preparations, billing, buying, commissions, profits and discounts pertinent to business transactions.
Geometry (2890 A 2900 B)
Grades 10-11
1 year Prerequisite:Completion of or enollment in Algebra 1
This course is designed as an introduction to Euclidian Geometry providing the necessary geometric concepts used in future mathematic courses. Topics include the study of patterns, lines, angles, transformations, triangles, polygons, solids, geometric figures, circles and deductive reasoning. Students willl be required to solve problems graphically, algebraically and symbolically. Use of a TI-83/84 calculator, nightly homework, and semester projects are required.
*Geometry (H) (2950 A 2960 B)
Grade 10
1 year Prerequisite: A 'C' average or better in Algebra I (H) or A "C" average in Algebra 1.
This course is designed for those students whose mathematics background indicates the ability to work at an advanced level. Students are expected to handle the Geometry cocepts at an increased depth of work and at a more rigorous pace. More emphasis is placed on informal proof of theorems as well as interraltaionships between Algebra and Geometry. Students will be required to solve problems graphically, algebraically, and symbolically. Use of a TI-83/84 calculator , nightly homework, and semester projects are required.
Algebra II (2830 A 2840 B) Grade 11
1 year Prerequisite: A 'C' or better average in Algebra I and Geometry and a teacher recommendation.
This course is designed as a further exploration in the study of algebraic concepts. Topics include: the studyof polynomial expressions, equations, inequalitities and functions with particular emphasis on quadratics. It also includes units on rational polynomials, radicals, complex numbers, matrices, conics, and logarithmic functions.Students will be required to solve problems graphically, algebraically, and symbolically. Use of a TI-83/84 calculator , nightly homework, and semester projects are required.
*Algebra II (H) (2860 A 2870
B) Grade 11
1 year Prerequisite: 'B' or better average in Algebra I and Geometry with a teacher recommendation.
This course is designed for those students whose mathematics background indicates the abilityto work at an advanced level. It is recommended for future math, science, or business majors. It should not be taken concurrently with geometry unless the student plans on taking Calculus during Senior year.Students are expected to handle Algebra II concepts at an increased depth and a more rigorous pace. Students will be required to solve problems graphically, algebraically, and symbolically. Use of a TI-83/84 calculator , nightly homework, and semester projects are required.
*$Intro College Algebra (3281 A 3282
B) Grade 12
1 semester or 1 year elective Recommended: College bound students who have below an 18 on the ACT and
are not enrolled in Advanced Math.
This course prepares those students for college algebra. Students have
the opportunity to receive a 'PASS' status for Developmental Mathematics at
Southeastern Louisiana University upon successful completion of this course.
*$Trigonometry /MA 162 (3150 B)
Grade 12
1 semester Prerequisite: Completion of Algebra II (H) and an 18 or above on the math section
of the ACT.
This course is designed for those students who plan to attend college upon graduation of high school. This course includes the study of trigonometric functions, graphing trigonometric functions, inverse trigonometric functions, function identities and the trigonometric triangle.Students will be required to solve problems graphically, algebraically, and symbolically. Use of a TI-83/84 calculator , nightly homework, and semester projects are required.
College Alg/MA 161 (3192 B)
Grade 12
1 semester Prerequisite: Completion of Algebra II(H) and an 18 or above on the math section
of the ACT with an overall composite score of 18. This course is designed for those students who plan to attend college upon graduation of high school. This course also includes the study of equations and inequalities, functions and their graphs, polynomial, rational, exponential and logarithmic functions. It also includes the study of systems of equations and inequalities and topics from analytic geometry. Students may earn three hours of credit at the university level upon successful completion of this course.Students will be required to solve problems graphically, algebraically, and symbolically. Use of a TI-83/84 calculator , nightly homework, and semester projects are required.
*Calculus (AP) (3111 A 3112
B)
Grades 11-12
1 year Prerequisite: Advanced math and trigonometry
Challenging course for students who excel in mathematics, and applications
will be explored. TI-83 calculator recommended. Small out-of-class
projects and nightly homework. This course prepares students to take
the AP calculus exam. |
Welcome to Mrs. Shea-LeTourneau's mathematics web page. This page will contain information about assignments, quizzes, and tests (please see the calendar). Along with helpful hints and resources.
Algebra 2 with Trig
This is a college prep course. Topics studied include: linear, quadratic, polynomial functions, systems of equalities and inequalities, rational and radical functions (introduced), exponential and logarithmic functions (introduced), series and sequences, and trigonometry (introduced). Students will be graded on daily assignments (25%), quizzes and tests (75%). One test retake is allowed per trimester under the following conditions: 1. all assignments must have been completed for the chapter, 2. student must come in for review/help, 3. the test and review must be completed within two weeks of the original test. |
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Giving Meaning to the Numbers
1/23/2007 - Jeff Marshall, Bob Horton, and Joyce Austin-Wade
When learning, students yearn for meaning, challenge, and relevance. Integrated learning fulfills these desires by limiting the compartmentalization of learning—providing a more coherent learning environment. Too often, mathematics and the physical sciences are taught as separate entities. Yet, many commonalities exist, especially between chemistry and Algebra II and between physics and precalculus (including trigonometry).
I (Jeff) am a science teacher, and my quest to integrate science and mathematics learning began over a decade ago when I participated in a two-year National Mathematics and Science Fellows project sponsored by the National Science Foundation and the Coalition for Essential Schools. As a science teacher, I welcomed the opportunity several years later when a math teacher also had a desire to integrate application-based learning. We created an integrated math/science course that specifically relates physics and precalculus concepts. I collaborated with my coauthor Bob to make sure the mathematics aligned with the most current mathematics standards and concepts.
Context
An alternative education program provided the context for this learning experience, but this course would also work in more traditional settings. In the featured alternative program, students take one or two courses at a time. Upon completion, students start the next course. For more traditional settings where classes are 55 minutes or 90 minutes long, students' classes can be blocked together (e.g., first-period physics and second-period mathematics or vice versa). Blocking student schedules provides longer exploration opportunities and flexibility to reconfigure groups so individual needs are met (e.g., a group of students can work on the computer to learn specific mathematics skills while another group gathers data to be analyzed).
Framework
The National Science Education Standards (NRC 1996) and the Principles and Standards for School Mathematics (NCTM 2000) share a significant number of standards. Figure 1 (p. 38) provides several examples where physics and mathematics standards logically parallel one another. Our integrated course unites recommendations from the College Board for Physics (2006), both the National Science and National Mathematics Standards, and the state standards. Although this is not an Advanced Placement class, being aware of the College Board's expectations helps lay the foundation for those taking a second year of physics.
Given a textbook and a list of standards, teachers too often start at the beginning of the textbook and "cover" as much material as possible until the year ends. To avoid this, our approach when developing the course began by planning backwards. First, we identified the skills and knowledge base in each discipline that students should attain by the end of the school year. Then, we condensed each discipline's list to the most critical things students should know and be able to do by the end of the course (McDonald 1992; Wiggins and McTighe 1998). Figure 2 (p. 39) details the list for physics and precalculus. The core of the integrated course focuses on commonalities and complementary ideas between the two courses. Standards that do not link well between disciplines are taught using an unlinked teaching approach. For instance, proving key trigonometric identities is a mathematical skill that does not have a parallel physics concept. We believe that trying to integrate marginally linked content would ultimately weaken the curriculum for one or both of the disciplines.
Because students receive credit in math and in physics, integrated assignments receive grades for both courses. Each teacher assesses different predetermined aspects of integrated assignments. For example, during inquiry investigations the math teacher places higher value on the data and results section, while the science teacher places greater emphasis on the ability to study a scientific problem, collect meaningful data, and draw accurate conclusions based on findings.
A tremendous benefit to an integrated math-science course is that redundancy is greatly reduced. This frees up time for exploring learning in greater depth or working longer on difficult-to-understand concepts. Additionally, the synergistic effect of working with individuals outside one's field can have a powerful impact on teaching and learning. In the past, students rarely made connections between physics and math; for example, using trigonometry in math class to study vectors in a coordinate plane was seen by students as different from applying vectors to two-dimensional motion problems in physics.
In the physics course, the integration of math concepts related to Newtonian mechanics (e.g., kinematics, motion, forces, and energy) receives the largest emphasis, followed by electricity and magnetism; waves and optics; and nuclear, thermodynamics, and modern physics. Trigonometric, polynomial, rational, exponential, and logarithmic functions receive the most attention in the mathematics course. The curriculum consists of various inquiry experiences, problem sets, computer instruction, quizzes to test concept mastery, small group instruction, projects, and a culminating portfolio. The final portfolio encompasses application of knowledge in seven different areas, such as graphing and motion, energy and momentum, and careers that use mathematics and physics.
I use full-class instruction infrequently in the science classroom—primarily to introduce a new unit, introduce major project assignments, and help pull major concepts and ideas together. Though inquiry-based instruction is becoming more common in mathematics classes, the math teacher uses full-class instruction—in particular direct instruction (Cruickshank, Jenkins, and Metcalf 2006)—much more frequently, as students learn and practice mathematics skills and concepts. Three specific examples of the integrated curriculum are detailed here.
Motion
Early in the year, students investigate linear motion—specifically, the concepts of displacement (distance), velocity, and acceleration. Guided- and open-inquiry labs (Martin-Hansen 2002) initially use stopwatches, metersticks, and graph paper for collecting data—thus making learning concrete and tangible. Computer simulations and calculator-based labs are added later to incorporate technology into learning experiences; in addition, they speed up the rate and accuracy of data collection.
One of the first motion labs provides students with 10 m of string, a timer, graph paper, a textbook, and a calculator; groups of four to five students prepare a report for the city council that answers whether speeding is a problem on the roads near the school. Because students conduct their investigation near a public road, parents sign a permission slip that allows their child to participate. Further, students are instructed to always stay a safe distance from the roadway. After discussing safety concerns, teams devise their methodology, which must be approved and signed by the teacher before starting. After collecting and analyzing the data, students give their response to the speeding issue. Visuals are later posted and referenced to help remind students of the meaning behind distance versus time or velocity versus time plots. Data analysis, problem-solving, and mathematical reasoning are also addressed by this lab.
In math class, statistics concepts—including mean, median, range, and standard deviation—now have purpose and meaning. Polynomial functions, including both linear and quadratic functions, are used for modeling the collected data. By exploring first- and second-order finite differences in the data, the seeds are sown for first- and second-order derivatives that occur in calculus, while giving specific meaning to slope.
Students must prepare presentations in which they defend why they use concepts such as mean or median or choose a particular function to model the speed of the cars. They quickly realize that one or two outliers (e.g., someone who is excessively speeding) can greatly skew the mean value. Finally, this investigation allows students to demonstrate Mastery of the Science as Inquiry standard that requires students to develop their abilities to do scientific inquiry as well as the process standards emphasized by National Council of Teachers of Mathematics (NCTM 1989, 2000).
Additionally, deeper conceptual understanding with other mathematical ideas is achieved when studying and graphing motion problems in laboratory investigations. Critical precalculus ideas such as rates and limits are clearly seen and applied. Investigating something that appears trivial, such as studying a bouncing ball, provides a concrete way for students to grasp the meaning associated with otherwise abstract concepts such as distance versus time, velocity versus time, or acceleration versus time graphs.
Overall, investigation of automobile speeds also helps transform students' understanding of mathematics as they learn to see math as a tool for modeling real-world phenomena. In absence of a concrete investigation such as this, slope merely becomes a rise over run algorithm to be completed. Limits become critical to the discussions and analyses as students are confronted with additional questions such as, "Is a bouncing ball always in motion?" and "What evidence supports your claims?" These discussions force students to extend beyond average rates of motion to address instantaneous rates of motion and understand that even a moving ball can have moments when its velocity is zero.
As students move on to calculus, they will be able to calculate more complex characteristics of motion using differentiation and integration. Integration allows a learner to calculate the area under the curved velocity versus time graph to determine the distance traveled. In precalculus settings, students use methods such as limits or summations of geometric shapes to calculate the areas under a curved velocity versus time graph.
Critical thinking
Four times throughout the course, specific critical-thinking investigations integrate mathematics and science. Each assignment has a different focus: logic and reasoning, science and math in everyday life, evidence and proof, and physics at the movies. In physics at the movies, the laws of physics are applied to action scenes from popular movies. Brief clips from some movies include the following (with applied concept): Chain Reaction (nuclear fusion), Michael Jordan's Playground (kinematics), Indiana Jones and the Last Crusade (impulse, momentum), Lord of the Flies (optics), 1492 (harmonic motion), The Fugitive (Newtonian mechanics), and Speed (Newtonian mechanics).
Mathematical functions are incorporated throughout. For example, using a clip from the movie 1492, students explore sinusoidal functions as they work with periodic motion. For another assignment students assume the role of a science correspondent for the 20th Century Fox movie studio for the making of the movie Speed. To make the movie more realistic, students draw and detail the conditions that would allow the bus to "safely" jump the 50 ft section of uncompleted highway. Additionally, using details provided in the movie, students use their knowledge of physics and mathematics to determine whether the bus could have made the jump; they must justify their conclusions and state all assumptions.
The jumping scene involves parabolic motion, which requires using physics concepts, algebraic functions, and trigonometric functions. Specifically, students must apply their knowledge of free-body diagrams that are central to any general physics class kinematics unit. Figure 3 diagrams the free-body layout for the Speed jump scene. Trigonometry is required to solve the resultant vectors needed to determine the conditions for a successful jump. For instance, the net force for the bus traveling on the ramp is the applied force less the friction force and the opposing component force of the gravitational force. The opposing component force of the gravitational force is calculated by mgsinÈ. These values reinforce triangle trigonometry. Students must approximate or find values for the following before they can solve the problem: launch angle, mass of the bus, and height of landing area in relation to launch point. This open-ended investigation integrates math and science content while allowing for unique solutions to be justified.
Figure 3. Diagram for the bus scene, modified from the movie Speed. [Note: Careful viewers of the film Speed have noted that the take off point for the bus was a flat section at the top of the bridge, making such a jump impossible (since the launch angle would be zero). The actual stunt was performed with a stunt bus, using a short take-off ramp hidden from view in the final editing of the film.]
Our setting allowed us to work with students collaboratively for three-hour blocks. Students were often not aware of when they were doing science or when they were doing mathematics. When math and science classes are taught separately, planning is essential to align the specific concepts and skills to be taught in both classes. In either situation, the skills that are being applied in physics class are then studied in greater depth to explore the underlying mathematical principles.
Solar racers
Miniature solar racers (Marshall 2004) provide an excellent opportunity to synthesize many of the concepts and ideas studied throughout the year in physics (motion, forces, electrical circuits, conservation of energy) with key mathematics content (trigonometric functions, accuracy, error, modeling, and problem solving). Although racing the mini solar cars provides motivation for learning, the majority of the learning transpires before the race even occurs.
For example, the power output of a solar array is directly correlated to the angle of incidence of the sun as it strikes the solar panel. When the sun is low in the sky such as early morning or late afternoon, the power output of the panel is greatly reduced when compared to the sun striking the panel perpendicularly. For instance, the power output is only half when the sun forms an angle of 30° with respect to the panel—assuming no other factors are involved such as partial cloud cover or reflections off other surfaces. This can be supported conceptually as well if you look at the area that a flashlight illuminates when shone directly overhead versus when it is lowered 60°. The larger circle shows that the light's intensity is more spread out when compared to the overhead position.
In the real world, situations are even more complex. To determine how long it takes a vehicle to get to the finish line, students test speed, accuracy, friction, different circuit combinations, and energy outputs. Variables such as angle of incidence of the light source, intensity of the light source, gear ratio used, mass of vehicle, and friction are critical factors in the overall success. The process of formulating and testing conjectures is just as important in math as it is in science. Specifically in math, students too often look at the dependent variable as a function of a single independent variable [y = f(x)]. Here students realize that the dependent variable is a function of several independent variables. This understanding is essential for students if they are to connect math class to the mathematics of the real world.
The race becomes a time to celebrate learning. Further, students quickly learn that the process and continuation of learning is valued over the product (the race results). After all, automotive corporations are successful after numerous prototypes have been tested—always learning from prior experience.
The reality
Strong relationships with colleagues, the willingness to plan with others, and flexibility are all requirements for successful integrated teaching. However, collaborative efforts are fraught with challenges. Working with counselors to schedule students is often a tricky endeavor, so administrative support is essential. Each school, program, and discipline will have its own challenges that will require creative solutions. If these challenges can be overcome, the reward becomes learning that more meaningfully applies to students' lives. The goal of integrated teaching and learning is realized when students begin questioning whether the current investigation is math or science.
If an entire integrated class is not feasible, coordinating one major concept per quarter with another discipline provides a great start. As you begin working with one or several teachers on making explicit connections between courses, the possibility of developing a more integrated curriculum becomes a reality. Find a math, history, literature, or art course that most of your students currently are enrolled in and seek to understand the standards and objectives for that course—you will be surprised the number of authentic connections that can be made with a little effort. When we willingly step beyond our own content areas, the meaning and relevance of learning becomes apparent for our students.
Jeff Marshall ([email protected]) is an assistant professor of science education and Bob Horton ([email protected]) is an associate professor of mathematics education, both at Clemson University in Clemson, South Carolina; Joyce Austin-Wade ([email protected]) is a mathematics teacher at Project HOPE in Oklahoma City, Oklahoma. |
Synopsis
"A review of basic arithmetic precedes clear explanations of how nurses need to apply mathematics in modern clinical practice. This study guide teaches an especially easy approach to solving the proportion problems key to converting medication orders and passing nursing licensing exams. The profusion of problems with detailed solutions, and hundreds more with answers, gives students ample opportunities to test their skills as they learn them--leading to quicker mastery. "
Found In
eBook Information
ISBN: 97800714258 |
This part of the tool allows the user to graph any surface in 3D and then plot the Taylor Polynomial of degrees 0 through 5. Both the algebraic equation and the graph are given of the Taylor Polynomial. The user can set the center point of the polynomial and can use a slider to change from degree to degree. CalcPlot3D also contains a parametric surface graphing capability, including the ability to display a "trace" point on the surface.
Discuss this resource
Ways you have used CalcPlot3D
by Paul Seeburger (posted: 11/14/2011 )
I would love to see some comments added to this resource to reflect the ways many instructors have used this applet in their teaching. I created this resource to help my students to see the connections between the concepts we study in multivariable calculus more clearly.
I use it to demonstrate certain properties in my lectures like the fact that the gradient vector points in the compass direction we should move along a surface in order to go most steeply uphill.
I also use it in lectures to visually verify solutions to various boardwork exercises like finding the equation of the plane determined by three non-collinear points and the intersection of two planes (or other surfaces).
I then get my students to use the applet by requiring them to visually verify solutions to various homework problems on graded worksheets. They print out their resulting graphs from the applet and hand them in with their homework. I have them do this with various topics including those mentioned above, as well as contour plots, level surfaces, tangent planes, flowlines through vector fields.
I also ask my students to complete several concept explorations that use the applet to consider the geometric properties of dot products, cross products, velocity and acceleration vectors, and Lagrange multiplier optimization.
I use it for some "what-if" types of explorations in class when we study space curves as well.
I would love to hear what others are doing, and in particular, I think it would be helpful to share particular functions, space curves, etc. and activities that could be done by students using this applet (or similar resources). |
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purpose is to promote and cultivate a higher curiosity, appreciation and
understanding of Mathematics for all the students at NIU. Also, Math Club
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Modern algebra is generally acknowledged to have begun with the appearance around 830 CE of al-Khwārizmī's book al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqābala. What better source can there be to find the answer to that perennial student question, "What is this stuff good for?" In the introduction to his seminal work, al-Khwārizmī stated that its purpose was to explain:
… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.
It doesn't get much more practical and useful than that! Either in 830 or today! Throughout history, the nations that led the world in mathematics led the world in commerce, industry, and science. In the 9th century, Baghdad was the commercial, industrial, and scientific center of world. In the 13th century, the leadership role crossed the Mediterranean to Italy, then over the ensuing centuries continued gradually westwards through Europe, crossing the Atlantic to the East coast of the US in the middle third of the 20th century, arriving in California in the 1980s, and likely to cross the Pacific (back) to China within the next couple of decades.
Al-Khwārizmī's strong emphasis on practical applications typified Arabic texts of the time, every bit as much as the intense focus on applications of mathematics and science you find in today's Silicon Valley.
The book was divided into three parts. The first part was devoted to algebra, giving the rules together with 39 worked problems, all abstract. Then came a short section on the rule of three and mensuration. Two mensuration problems dealing with surveying were solved with algebra. Finally, al-Khwārizmī presented a long section on inheritance problems solved by algebra.
The term al-jabr ("restoration" or "completion") in al-Khwārizmī's title refers to a procedure whose modern counterpart is eliminating negative terms from a (linear or quadratic) equation by adding an appropriate qantity to both sides of the equation. For example, using one of al-Khwārizmī's own examples (but expressed using modern symbolic notation), al-jabr transforms
x2 = 40x – 4x2
into
5x2 = 40x.
The other key term in the title, al-muqābala ("confrontation") refers to the process of eliminating identical quantities from the two sides of the equation. For example, (again in modern notation) one application of al-muqābala simplifies
50 + 3x + x2 = 29 + 10x
to
21 + 3x + x2 = 10x
and a second application simplifies that to
21 + x2 = 7x.
Procedurally (but not conceptually) these are the methods we use today to simplify and solve equations. Hence, a meaningful, modern English translation for Hisâb al-Jabr wa'l-Muqābala would be, simply, "Calculation with Algebra."
The symbolic notation is not the only difference between medieval algebra and its present-day counterpart. The medieval mathematicians did not acknowledge negative numbers. For instance, they viewed "ten and a thing" (10 + x) as a composite expression (it entails two types of number: "simple numbers" and "roots"), but they did not see "ten less a thing" (10 – x) as composite. Rather, they thought of it as a single quantity, a "diminished" 10, or a 10 with a "defect" of x. The 10 retained its identity, even though x had been taken away from it. When an x was added to both sides of an equation, the diminished 10, (10 – x), was restored to its rightful value. Hence the terminology.
The first degree unknown, our x, was usually called shay' ("thing"), but occasionally jidhr ("origin" or "base", also "root" of a tree, giving rise to our present-day expression "root of an equation"). The second power, our x2, was called māl (a sum of money/property/ wealth). Units were generally counted in dirhams, a denomination of silver coin, occasionally simply "in number". For example, al-Khwārizmī's (rhetorical) equation "a hundred ten and two māls less twenty-two things equals fifty-four dirhams" corresponds to our symbolic equation 110 + 2x2 – 22x = 54.
Arabic authors typically explained the methods of algebra in two stages. First they provided an explanation of the names of the powers, described six simplified forms of equations and their solutions, and gave rules for operating on polynomials and roots. They then followed this introduction by a collection of solved problems which illustrated the methods.
Their solutions followed a standard template:
Stage 1: an unknown quantity was named (usually referred to as a "thing"), and an equation was set up.
Stage 2: the equation was simplifed to one of six canonical types.
Stage 3: the appropriate procedure was applied to arrive at the answer.
Because they allowed only positive coefficients, they had to consider six equation types, rather than the single template ax2 + bx + c = 0 we use today:
(1) māls equals roots (in modern terms, ax2 = bx),
(2) māls equals numbers (ax2 = c),
(3) roots equals numbers (bx = c),
(4) māls and roots equals numbers (ax2 + bx = c),
(5) māls and numbers equals roots (ax2 + c = bx),
(6) māls equals roots and numbers (ax2 = bx + c).
We see how al-Khwārizmī used the two simplification steps in Stage 2, al-jabr wa'l-muqābala, ("restoration and confrontation") in his solution to a quadratic equation, which he described in these words:
.
The American scholar Jeffrey Oaks has translated this (fairly literally) as follows, adding headings to assist the reader:
Enunciation If [someone] said, ten: you divided it into two parts. You multiplied one of the parts by itself, which is the same as eighty-one times the other.
Setting up and simplifying the equation The rule for this is that you say ten less a thing by itself is a hundred and a mal less twenty things [which] equal eighty-one things. Restore the hundred and a mal by the twenty things and add them to the eighty-one [things]. This yields: a hundred and a mal equal a hundred roots and a root.
Solving the simplified equation So halve the roots, which yields fifty and a half, and multiply it by itself, which yields two thousand five hundred fifth and a fourth. Subtract from it the hundred, leaving two thousand four hundred fifty and a fourth. Take its [square] root, which is forty-nine and a half. Subtract it from half the roots, which is fifty and a half. There remains one, which is one of the two parts.
Using modern notation, and substituting the letter x for "thing", al-Khwārizmī was solving the equation
(10 − x)2 = 81x
which can be written in the equivalent form
x2 + 100 = 101x
Al-Khwārizmī did not state the equation
(10 − x)2 = 81x
Rather, he set up the equation
100 + x2 – 20x = 81x.
Nothing like the equation (10 − x)2 = 81x was ever stated in medieval algebra; the left side of such an expression entails what was then an unrealized operation. Medieval algebraists worked out all operations before stating equations, so al-Khwārizmī did not begin with (10 − x)2 = 81x, as we would, rather he first worked out the multiplication.
Having demonstrated methods for solving linear and quadratic equations, al-Khwārizmī proceeded to examine how to manipulate algebraic expressions. For example he showed how to multiply out specific numerical instances
(a + bx) (c + dx)
expressing everything in words, not symbols.
He ended the first section of the book by presenting the solutions to 39 problems.
In the following section, al-Khwārizmī presented solutions to some mensuration problems, including rules for finding the area of figures such as the circle and for finding the volume of solids such as the sphere, cone, and pyramid.
The final part of the book dealt with the complicated Islamic rules for inheritance, which involved the solution of linear equations.
* * *
COMING UP: In the final article in this series I'll summarize some of the amazing developments in algebra that were made in the Arabic period subsequent to al-Khwārizmī.
Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c.780 – c.850 CE) was one of the most significant figures in the development of modern algebra. Yet we know virtually nothing about his life.
There is even some confusion in the literature as to his full name. Most present-day sources give it as Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, which can be translated as "Father of ʿAbdallāh, Mohammed, son of Moses, native of the town of al-Khwārizmī". References to Abū Jaʿfar Muḥammad ibn Mūsā al-Khwārizmī are erroneous in this context; that was a different person
Al-Khwārizmī wrote several books, two of which had a huge impact on the growth of mathematics, one focused on arithmetic, the other on algebra. He aimed both at a much wider audience than just his fellow scholars. As with Euclid and his Elements, it is not clear whether al-Khwārizmī himself developed some of the methods he desribed in his books, in addition to gathering together the work of others, though a later author, Abū Kāmil, suggested that his famous predecessor did develop some of the methods he presented in his books.
The first of al-Khwārizmī's two most significant books, written around 825, described Hindu-Arabic arithmetic. Its original title is not known, and it may not have had one. No original Arabic manuscripts exist, and the work survives only through a Latin translation, which was most likely made in the 12th century by Adelard of Bath. The original Latin translation did not have a title either, but the Italian bibliophile Baldassare Boncompagni gave it one when he published a printed edition in the 19th century: Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning"). The Latinized version of al-Khwārizmī's name in this title (Algoritmi) gave rise to our modern word "algorithm" for a set of rules specifying a calculation. In English, the work is sometimes referenced as On the Calculation with Hindu Numerals, but it is most commonly referred to simply as "al-Khwārizmī's Arithmetic."
Al-Khwārizmī's second pivotal book, completed around 830, was or more loosely as "reducing (or solving) an equation." The title of the book translates literally as "The Abridged Book on Calculation by Restoration and Confrontation", but a more colloquial rendering would thus be "The Abridged Book on Algebra". It is an early treatise on what we now call "algebra," that name coming from the term al-jabr in the title. Scholars today usually refer to this book simply as "Al-Khwārizmī's Algebra." There are seven Arabic manuscripts known, not all complete. One complete Arabic copy is kept at Oxford and a Latin translation is kept in Cambridge. Two copies are in Afghanistan.
In Algebra, al-Khwārizmī described (but did not himself develop) a systematic approach to solving linear and quadratic equations, providing a comprehensive account of solving polynomial equations up to the second degree.
The Algebra was translated into Latin by Robert of Chester in 1145, by Gherardo of Cremona around 1170, and by Guglielmo de Lunis around 1250. In 1831, Frederic Rosen published an English language translation. In his preface, Rosen wrote:
ABU ABDALLAH MOHAMMED BEN MUSA, of Khowarezm, who it appears, from his preface, wrote this Treatise at the command of the Caliph AL MAMUN, was for a long time considered as the original inventor of Algebra. … … … From the manner in which our author [al-Khwārizmī], in his preface, speaks of the task he had undertaken, we cannot infer that he claimed to be the inventor. He says that the Caliph AL MAMUN encouraged him to write a popular work on Algebra: an expression which would seem to imply that other treatises were then already extant.
In fact, algebra (as al-Khwārizmī described it in his book) was being transmitted orally and being used by people in their jobs before he or anyone else started to write it down. Several authors wrote books on algebra during the ninth century besides al-Khwārizmī, all having the virtually identical title Kitāb al-ğabr wa-l-muqābala. Among them were Abū Hanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Tūsī.
In addition to his two books on mathematics, al-Khwārizmī wrote a revised and completed version of Ptolemy's Geography, consisting of a general introduction followed by a list of 2,402 coordinates of cities and other geographical features. Titled Kitāb ṣūrat al-Arḍ("Book on the appearance of the Earth" or "The image of the Earth"), he finished it in 833. There is only one surviving Arabic copy, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.
* * *
COMING UP NEXT: Al-Khwārizmī's answer to that perennial student question, "What is algebra good for?" Plus a look at the contents of his seminal book, including an explanation of what exactly was being "restored" in the process for which al-Khwārizmī's Arabic term was al-jabr.
* * *
Al-Khwārizmī on National Public Radio: I talked about al-Khwārizmī and the birth of algebra with host Scott Simon in my occasional "Math Guy" slot on NPR's Weekend Edition on December 24.
On 14 September 786,Harun al-Rashid became the fifth Caliph of the Abbasid dynasty. From his court in the capital city of Baghdad, Harun ruled over the vast Islamic empire, stretching from the Mediterranean to India. He brought culture into his court and encouraged the widespread pursuit of learning.
Al-Rashid had two sons, the elder al-Amin, the younger al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad.
Al-Mamun continued the patronage of learning started by his father. With his encouragement, scholars of the time set about collecting and writing down in books all available practical knowledge, much of which had hitherto been transmitted only orally, including mathematics and folk astronomy. They translated into Arabic works of Greek and Indian science.
Many of the works collected and created may have been housed in a library called the House of Wisdom, though there is no evidence to support the commonly repeated claims that (1) it was massive, (2) it was founded by al-Mamun, or (3) translations were carried out there.
The tradition of learning, writing, and translation begun by al-Rashid and al-Mamun continued for the next quarter century, making the Islamic civilization the center of world knowledge. The aristocracy and other wealthy groups within Muslim society supported the appropriation of all practical and scientific knowledge they could acquire. They employed scholars to translate into Arabic works by Indian, Sasanian, and especially Greek authors, and mathematicians recorded on paper all that was known of arithmetic, algebra, and mensuration, which had hitherto been communicated orally by traders. In addition to the mathematical sciences (arithmetic, geometry, optics, mathematical astronomy, etc.), they also translated texts on geography, astrology, philosophy, medicine, agriculture, alchemy, and even falconry.
Greek works formed the bulk of the material translated. In addition, the more scientifically oriented mathematicians adopted the Greek tradition of definitions, axioms, and propositions with rigorous proof, and astronomers embraced the Greek idea of geometric models of planetary motion. Within this framework, Indian techniques were incorporated into this new Arabic/Islamic mathematics.
In addition to the translations, scholars wrote commentaries and criticisms of the ancient mathematics and made their own original contributions. For example, in the 9th century, Thābit ibn Qurra (d. 901) translated several works of Archimedes, wrote commentaries on Euclid's Elements and Ptolemy's Almagest, critiqued Euclid's definition for the composition of ratios of numbers, and derived and proved new formulas for volumes of solids of revolution.
When the sources of Greek and other foreign texts was finally exhausted, scholars continued to produce new results in all branches of mathematics. For instance, in the 11th century, Ibn al-Haytham made major contributions to optics and geometry, and at the start of the 12th century, al-Khāyyamī wrote his book on algebra.
Over a thousand mathematical manuscripts from the period have survived, about half of them dating before the 15th century.
Al-Khwārizmī, who may have studied and worked in the House of Wisdom, was one of the earliest contibutors to this vast undertaking, and arguably had the most impact of all the mathematicians involved. But his books – he wrote one on Hindu arithmetic in addition to the one on algebra – should be viewed as part of this larger movement.
At the time, algebra was viewed primarily as a practical, numerical problem solving technique, not the autonomous branch of mathematics it became later. Indeed, the greatest contribution of Arabic mathematical work to society was its development as a set of practical tools.
Three systems of practical calculation were taught and practiced in the medieval Islamic world: finger reckoning, Hindu arithmetic, and the base 60 system of the astronomers. Merchants preferred finger-reckoning, which worked for numbers up to 10,000. Finger reckoning was used to solve problems by various methods, such as double false position and algebra. Al-Khwārizmī is known to have written a work, now lost, called Book of Adding and Subtracting, in the early 9th century, which was probably devoted to the use of finger reckoning. (If so, it was probably the earliest written text on the subject.)
The Arabic mathematicians referred to the numerals 1, 2, 3, etc., as Hindī numerals, because they acquired the system from India. These numerals were already in use in the Middle East by the 7th century CE. The earliest known Arabic text describing the system is al-Khwārizmī's Book on Hindī Reckoning, written in the early 9th century, which survives only in Latin translation. The original algorithms for calculating in this system were devised for use on a dust board, where erasing is easy. In the middle of the 10th century, al-Uqlīdisī introduced new algorithms for use with pen and paper. The Arabic mathematicians introduced the concept of decimal fractions, wihich al-Uqlīdisī described for the first time.
Unlike Diophantus, most of the Arabic authors, including al-Khwārizmī, wrote their algebra almost entirely in words. For example, where we would write down the symbolic equation x + 1 = 2, they might write "The thing plus one equals two" (and very occasionally "The thing plus 1 equals 2"). This is generally known as the rhetorical form, and remained in common use right up to the 16th century. This is, however, a notational distinction, not one of content. Commentators who refer to "rhetorical algebra" as being a form of algebra distinct from "literal algebra" are in error. For, although the Arabic authors wrote their books rhetorically, with no notation even for numbers, they did not solve problems rhetorically. Throughout most of Arabic algebra, problems were worked out on some ephemeral surface, by writing the coefficients and numbers in Hindu form. For example, they would write
1 2 1
to mean x2 + 2x + 1. Later, Arabic scholars in the Maghreb developed a truly algebraic notation, with symbols for the words representing the powers of the unknown, but even they they would resort to rhetorical text to communicate the result of a calculation.
Symbolic algebra, where full symbolism is used, is generally credited in the first instance to the French mathematician François Viète (1540 –1603), followed by René Descartes (1596 – 1650), though traces can be discerned in the writings of some Arabic mathematicians as early as the 13th century.
* * *
In my next two articles in this short series, I'll say a bit about al-Khwārizmī and take a look at the contents of his seminal book on algebra. In particular, I'll give his answer to that perennial student question, "What is algebra good for?"
I use terms like "Arabic mathematics" in the standard historical fashion to refer to the mathematics done where and when the primary language for scholastic texts was Arabic. Mathematics, like all of science, belongs to the world.
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The second in a series. See the November 20 entry, "What is algebra?" for the first.
Two key features of algebra as we understand the word today are:
1. Reasoning about numbers by recognizing patterns across numbers;
2. Solving a problem by introducing a term for an unknown and then, starting with what is known, reasoning to determine its value.
We first see the emergence of both features of algebra in the mathematics of ancient Babylonia, around 2,000 BCE.
Several hundred of the many thousands of Babylonian's cuneiform-inscribed clay tablets that have been found are devoted to mathematics. They show that those ancient mathematicians had systematic procedures for solving geometric problems involving the determination of lengths and areas of figures. Today, we would solve those kinds of problems using linear and quadratic equations and indeterminate systems of linear equations. Their methods amounted to a form of geometric algebra that could be applied to solve problems beyond overtly geometric examples such as calculating the perimeters or areas of various plane figures or the volumes of solid objects: arithmetic problems arising in trade and commerce, for example, and other financial transactions such as inheritance. In addition, the Babylonians considered problems that seemed to have had no practical application, pursuing them purely for recreation. Although they described their procedures in terms of specific lengths and areas, they did so in a way that made it clear they applied in general, and in that sense they were starting to think algebraically, by recognizing patterns across quantities.
Moreover, some of their writings show the second characteristic feature of algebra, namely introducing an unknown and then reasoning to find its value. In their case, however, the unknown was not numeric but geometric – an unknown line on which they performed geometrical operations to get the answer.
In reasoning with unknown quantities, the Babylonians went further than other early civilizations with a mathematical tradition, such as the Egyptians, the Chinese, and the early Greeks, all of the first millennium BCE. Our knowledge of the mathematics of those peoples comes from works such as the Rhind papyrus, The Nine Chapters of the Mathematical Art, and Euclid's Elements, respectively. The approach described in those documents was, like that of the Babylonians, fundamentally geometric and exhibited reasoning about patterns of quantities, but we do not find the introduction of an unknown followed by an argument to determine its value.
It is with the work of the Greek mathematician Diophantus (ca. 210–290 CE) that we first find clearly recognizable algebra, where the unknowns represent numbers whose values are to be determined. Around 250 CE, Diophantus, who lived in Alexandria in Egypt, wrote a multi-volume work, Arithmetica, which its title notwithstanding was an algebra book. Its author used letters (literals) to denote the unknowns and to express equations, but that is a purely notational distinction. He also was one of the first mathematicians to use negative numbers in calculations. He showed how to solve equations by using two techniques called restorationand confrontation. In modern terms, these correspond more or less (but not precisely)to (1) adding a quantity to both sides of an equation to eliminate a negative term on one side, and (2) eliminating like terms from both sides. He used these techniques to solve polynomial equations involving powers up to 6.
Almost four hundred years later, the Indian mathematician Brahmagupta (598–668 CE) likewise displayed recognizable algebra, in his book Brahmasphutasiddhanta, where he described the first complete arithmetic solution (including zero and negative solutions) to quadratic equations.
Following Diophantus and Brahmagupta, the next major step in the development of algebra – and it was huge – took place in the period generally referred to as "Arabic mathematics" or "Muslim mathematics", a significant outpouring of mathematical activity stretching from the 8th century to the end of the 16th. Indeed, the word algebra itself comes from the Arabic word al-jabr, which occurs in the title of a highly influential book by the Persian mathematician al-Khwārizmī, completed around 830: but more loosely means "solving an equation."
That period will be the focus of my next article on algebra.
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We hear a lot about the importance that all children master algebra before they graduate from high school. But what exactly is algebra, and is it really as important as everyone claims? And why do so many people find it hard to learn?
Answering these questions turns out to be a lot easier than, well, answering a typical school algebra question, yet surprisingly, few people can give good answers.
First of all, algebra is not "arithmetic with letters." At the most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues. (I should stress that in this article I'm focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)
Let's start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today's Iraq) around 10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money as a means to measure an individual's wealth and mediate the exchange of goods and services. The monetary tokens eventually gave way to abstract markings on clay tablets, which we recognize today as the first numerals (symbols for numbers). Over time, those symbols acquired an abstract meaning of their own: numbers. In other words, numbers first arose as money, and arithmetic as a means to use money in trade.
It should be noticed that counting predates numbers and arithmetic by many thousands of years. Humans started to count things (most likely family members, animals, seasons, possessions, etc.) at least 35,000 years ago, as evidenced by the discovery of bones with tally marks on them, which anthropologists conclude were notched to provide what we would today call a numerical record. But those early humans did not have numbers, nor is there any evidence of any kind of arithmetic. The tally markers themselves were the record; the marks referred directly to things in the world, not to abstract numbers.
Something else to note is that arithmetic does not have to be done by the manipulation of symbols, the way we are taught today. The modern approach was developed over many centuries, starting in India in the early half of the First Millennium, adopted by the Arabic speaking traders in the second half of the Millennium, and then transported to Europe in the 13th Century. (Hence its present-day name "Hindu-Arabic arithmetic.") Prior to the adoption of symbol-based, Hindu-Arabic arithmetic, traders performed their calculations using a sophisticated system of finger counting or a counting board (a board with lines ruled on it on which small pebbles were moved around). Arithmetic instruction books described how to calculate using words, right up to the 15th Century, when symbol manipulation began to take over.
Many people find arithmetic hard to learn, but most of us succeed, or at least pass the tests, provided we put in enough practice. What makes it possible to learn arithmetic is that the basic building blocks of the subject, numbers, arise naturally in the world around us, when we count things, measure things, buy things, make things, use the telephone, go to the bank, check the baseball scores, etc. Numbers may be abstract — you never saw, felt, heard, or smelled the number 3 — but they are tied closely to all the concrete things in the world we live in.
With algebra, however, you are one more step removed from the everyday world. Those x's and y's that you have to learn to deal with in algebra denote numbers, but usually numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction. Doing so requires quite a lot of effort and training.
The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking. Those formulas and equations, involving all those x's and y's, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music. In fact, traders and other people who needed it used algebra for 3,000 years before the symbolic form was introduced in the 16th Century. (That earlier way of doing algebra is nowadays referred to as "rhetorical algebra," to distinguish it from the symbolic approach common today.)
There are several ways to come to an understanding of the difference between arithmetic and (school) algebra.
First, algebra involves thinking logically rather than numerically.
In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
In arithmetic, you calculate a number by working with the numbers you are given; in algebra, you introduce a term for an unknown number and reason logically to determine its value.
The above distinctions should make it clear that algebra is not doing arithmetic with one or more letters denoting numbers, known or unknown.
For example, putting numerical values for a, b, c in the familiar formula
in order to find the numerical solutions to the quadratic equation
is not algebra, it is arithmetic.
In contrast, deriving that formula in the first place is algebra. So too is solving a quadratic equation not by the formula but by the standard method of "completing the square" and factoring.
When students start to learn algebra, they inevitably try to solve problems by arithmetical thinking. That's a natural thing to do, given all the effort they have put into mastering arithmetic, and at first, when the algebra problems they meet are particularly simple (that's the teacher's classification as "simple"), this approach works.
In fact, the stronger a student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For example, many students can solve the quadratic equation x2 = 2x + 15 using basic arithmetic, using no algebra at all.
Paradoxically, or so it may seem, however, those better students may find it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to stop thinking arithmetically and learn to think algebraically.
Is mastery of algebra (i.e., algebraic thinking) worth the effort? You bet — though you'd be hard pressed to reach that conclusion based on what you will find in most school algebra textbooks. In today's world, most of us really do need to master algebraic thinking. In particular, you need to use algebraic thinking if you want to write a macro to calculate the cells in a spreadsheet like Microsoft Excel. This one example alone makes it clear why algebra, and not arithmetic, should now be the main goal of school mathematics instruction. With a spreadsheet, you don't need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can. What you, the person, have to do is create that spreadsheet in the first place. The computer can't do that for you.
It doesn't matter whether the spreadsheet is for calculating scores in a sporting competition, keeping track of your finances, running a business or a club, or figuring out the best way to equip your character in World of Warcraft, you need to think algebraically to set it up to do what you want. That means thinking about or across numbers in general, rather than in terms of (specific) numbers.
Of course, the need for algebra does not make it any easier to learn — though I think that spreadsheets can provide today's students with more meaningful and fulfilling applications than problems about trains leaving stations or garden hoses filling swimming pools, that my generation had to endure. But in a world where our very national livelihood depends on staying ahead of the technology curve, it is crucial that we equip our students with the kind of thinking skills today's world requires. Being able to use computers is one of those skills. And being able to use a computer to do arithmetic requires algebraic thinking.
In future postings I'll describe the growth of algebra through the ages.
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The fact that I, along with millions of other people, learned of Steve Jobs' death by way of a text message or email sent to my iPhone indicates just how huge was the Apple co-founder and CEO's impact on the way many of us go about our business and live our lives. I can't imagine any other CEO of a large corporation whose death would be so widely mourned.
By chance, the Apple Store in Job's home town of Palo Alto, where I live, is right across the street (University Avenue) from the Peet's Coffee Shop where I go for my morning latte. As I left the coffee house the morning after his death was announced, I noticed some flowers and a couple of lighted candled had been left on the sidewalk in front of the Apple Store window. I walked over and took a couple of photos. A day later, that small shrine had grown considerably.
Since Jobs had lived less than a mile from my home, on the Saturday morning I detoured past his (surprisingly modest) suburban-style house on my way to the supermarket, where I found precisely what I had anticipated. A huge outpouring of grief and remembrances.
Again it struck me: this reaction to the passing of a billionaire CEO? I cannot imagine a similar response to the death of Bill Gates, Larry Page, Sergey Brin, Mark Zuckerberg, or any of the other technology elite. Somehow, Jobs did more than provide people with useful consumer products, he touched their emotions.
To be sure, not everyone is an Apple fan. But to those who are, their attachment to the brand is deep. People tend not to like Apple stuff, rather they LOVE it. And that is surely a tribute to superb design – design both for appearance and for usability.
I can't say I've ever been a fan of anyone or anything, but I am a sucker for great design. When I first saw, and used, a Macintosh (one of the first generation), it quite literally changed my career, and in due course my life. As a mathematician with research interests in computation and a side interest (more of a hobby) in computational number theory, I was using computers long before the personal computer came on the scene. Back then (and this was the 1970s), a computer was a slave that you commanded to do things for you. Even worse, it was a dumb slave, so you had to formulate those commands with extreme care. A single missing or misplaced comma would cause the machine to grind to a halt.
All that changed with the Mac. As a user, you were no longer issuing commands to be acted upon, you were HANDLING INFORMATION. It was your stuff, and you were the one performing the action. Or so it seemed. In reality, you were being fooled by a cleverly designed interface. But that was the genius of the Mac; it took something intrinsically alien to human beings, computation, and presented it is a way that we find natural and instinctive.
Well, actually, that genius was not Apple's, rather that of a remarkable group of researchers at the Xerox Palo Alto Research Center (PARC) who had developed this new approach to computing over several years. To Xerox (and many of the researchers at PARC), the goal was to build computers to support office workers. Jobs' genius was recognizing that, properly packaged and marketed, this approach to computing could turn computers into mass-market consumer products.
When I first started to use a Mac, I realized that there was huge value to be had in viewing reasoning not as a PROCESS of logical deduction to arrive at a conclusion (the classical view of mathematical reasoning), but as GATHERING INFORMATION in order to reach a decision. The result of that shift in viewpoint first resulted in my book Logic and Information, which was published in 1990, and my entire research career since then has followed on the heels of that book.
Were it not for the Mac, I doubt I would have shifted my research the way I did. It would not have been enough to read about the WIMPS interface (windows, icons, mouse, pointers), or even to watch someone else using it. It was the powerful sensation of DOING it yourself, of physically manipulating items of information, that made all the difference. The Mac was not a device you used. It was something you experienced. And that is a profound difference.
As became clear through his entire career, Jobs had a deep appreciation for the importance of making technology something people experience – not merely use.
By pure chance, just two months before Jobs died, I published a short e-book in which I compared him with one of the greatest innovators of all time, the thirteenth century mathematician Leonardo of Pisa, known more commonly today by the modern nickname Fibonacci. Leonardo changed the world by writing a book that introduced modern arithmetic to the western world.
The stories of what these two men did exhibit remarkable similarities. Leonardo packaged arithmetic and, through the medium of parchment, gave personal computing to the masses. Jobs made personal computing accessible to everyone through the medium of silicon. Neither individual was an inventor. Their genius was taking something alien and complex and making it accessible – and friendly – to all. |
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Mathematical Learning Aids
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The way "The Algebra Toolbox" powerpoints are written shows the author has an excellent understanding of what students find difficult about algebra and how they commonly get confused. This means that students could be allowed to work through "The Algebra Toolbox" at their own pace, knowing that their usual misunderstandings will be addressed, generally before they get them!
Queensland Association of Mathematics Teachers Journal
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The Algebra Toolbox is a mathematical learning tool for Grade 7 to Grade 12. Comprising over 150 topic-based mathematics PowerPoints the Algebra Toolbox has had phenomenal success around the world most notably in Australia, The United Kingdom and New Zealand.
Prepared by a mathematics teacher with over 30 years' experience, the PowerPoints are renowned for their presentation, quality and clarity of explanation. They are ideal for… |
Maths should be a compulsory subject for all students after the age of 16, the House of Lords Science and Technology Committee says.
A report by the committee has found that many students starting STEM (science, technology, engineering and maths) degrees, even those with A-Level maths qualifications, lack the math skills required to undertake their studies.
It has called for all students to continue studying maths past the age of 16, and for all students who want to study STEM at university to study the subject to A-level standard. It has also called for universities to toughen up their maths requirements for entry into STEM courses and get more involved in setting up the maths curriculum.
Lord Willis, chairman of the Lords sub-committee on higher education in STEM subjects said: "We were absolutely gobsmacked that 20 per cent of engineering undergraduates do not have A2 (A-level) mathematics, 38 per cent of chemistry and economics undergraduates do not have A2 maths and 70 per cent of biology undergraduates do not have A-level maths."
"If we are talking about a world-class STEM base, where mathematics is the cornerstone of virtually every science programme, then it is really quite amazing that we have so few students who have studied maths, literally, beyond GCSE and often, not even with a grade A."
Professor Sir William Wakeham, international secretary and senior vice-president of the Royal Academy of Engineering, who was the specialist adviser for the committee said they had spoken to pharmaceutical industries who have "enormous demand" for statistical analysis on the effects of their drugs.
Many of their graduates have studied biological science and "not studied maths from the age of 16 with a minimal level of statistics", he said.
"Employers are rather keen that all of their students should have these kinds of skills," he said.
Sir William added that because of the modularisation of exams, it is possible "to avoid whole subjects in maths, like calculus and still find yourself in an engineering discipline where maths is essential".
Lord Willis said there were some engineering students that had "virtually no understanding" of mechanics.
A number of university vice-chancellors told the sub-committee that their institution was being forced to offer remedial maths classes not only for those that had not studied the subject at A-level, but for those who had taken it and done well, the report said.
Professor Sir Christopher Snowden, vice-chancellor of Surrey University, told the group: "I think that in pretty much every university the issues over maths skills apply.
"Indeed, this has been an issue now for many years within universities, partly due to the increase in the breadth of maths that is studied at schools but with a lack of depth.
"In some cases, for example, there is a complete absence of calculus, which is an issue in many subjects."
The sub-committee has recommended that the Government should make maths compulsory for all students after GCSE. "We share the view that all students should study some form of maths post-16, the particular area of maths depending on the needs of the student.
"For example, prospective engineering students would require mechanics as part of their post-16 maths, whereas prospective biology students would benefit from studying statistics."
It adds: "We recommend also that maths to A2 level should be a requirement for students intending to study STEM subjects in higher education."
The report also considered the recent changes to the UK's immigration rules and said it had led to a perception that the UK did not welcome students.
Lord Willis said: "Combined with the increases in tuition fees, this risks damaging universities funding base and limiting their ability to offer higher quality STEM courses. The Government must take steps to ensure international students are not put off studying here."
Gareth James, head of education at the Institution of Engineering and Technology (IET) said: "From across industry we hear repeated time and again that there are not enough young people with the right qualifications available to take the rewarding and challenging engineering careers that are available now and anticipated over the next couple of decades.
"Good quality qualifications in maths, physics, chemistry, biology and computer science will open many doors to young people but if they want to stand out as potential employees they also need to demonstrate their ability to apply their learning and to have good employability skills such as team working. It is vital that if young people want to have the best chance in the jobs market that they need to research what employers are looking for, to choose their subjects accordingly and gain the sort of experience and skills that will make them appealing."
The Lords Science and Technology Committee is right to call for urgent action by the government to boost student numbers in these subjects, the IET said.
A Department for Education spokeswoman said: "We want the majority of young people to continue studying maths up to 18 to meet the growing demand for employees with maths skills.
"We are reviewing how maths is taught in schools and overhauling GCSEs and A-levels to make sure they are robust and in line with the best education systems in the world." |
GrafEq
GrafEq is program for producing graphs of implicit equations and inequalities. Complete integration into the high-school curriculum is possible with site licenses. GrafEq is designed to foster a visual understanding of mathematics by providing a reliable graphing engine that encourages mathematical exploration. GrafEq is used in schools and homes and offers English, Dutch, Spanish, French, Brazilian Portuguese, Indonesian, Korean, and Japanese interface.
Price
$60.00
License
Free to try
File Size
1.36 MB
Version
2.13 |
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Build Intellectual Foundations
Quantitative and Mathematical Reasoning
These courses prepare you to understand the complexities of the modern world by ensuring a basic ability to reason mathematically (broadly defined), and an appreciation for more advanced mathematical thinking and analysis. |
Precalculus -- Preparing Students for
Calculus
Precalculus,
by Warren Esty
Sixth edition (This page updated Sept. 25, 2012)
A text
designed to produce a deep understanding of algebra and
trigonometry so that students will be comfortable with their
next math. Students will be well-prepared for calculus.
This text is designed to be appropriate for self-study, as
well as classroom use.
The content includes the usual precalculus
material (functions, powers, polynomials, logs, exponentials,
trig, etc.). Graphing calculators play an important role.
However, this text is unlike others because it does not just
use calculators to do old-style problems, but actually
incorporates calculators as a learning tool
and not just a "doing" tool.
This text has been used at Montana State University and
elsewhere by about a hundred different instructors and many
thousands of students. A great deal of experience has gone into
making this text an effective learning tool.
"As a home-schooled junior in high school
this year, I used your Precalculus book. [clip] I would just
like to thank you so much for writing that book. It changed my
entire perspective on math. Up to this year, I considered math
to be a distasteful medicine [clip]. No text book on a language
was ever written better. I quickly became enthralled with your
ideas, and now I love Math. Precalculus was my favorite subject
this year [clip. Read the whole letter
here.]
"I ordered your precalculus book, and thus far
it's been quite a learning experience. I've learned more in the
past 3 months than I learned in 4 years of high school. ....
Thanks." [an adult student]
"I'm taking Precalculus for graduate school,
and I have always hated math and been terrible at it, I was taught
only to memorize, never to understand. I'm starting to love math,
and it's because my dad (who loves math) has your book and we've
been working with it! So I wanted my own copy. Thank you for
writing such a great precalculus book!"
"I am the teacher who used your
Precalculus text with some homeschoolers over the internet the
year before last. I now have more students asking for the class.
I would like to use your text again. (I was very pleased that
the students who used your book and went on to take the SAT got
math scores very near 800. It'll be interesting to see what
happens with the new SAT exam.) ..."
"I love teaching the text and the other
instructor feels likewise. The course has been a challenge for my
students. For those who have buckled
down and done the requisite assigned work (and shift in conceptual
focus), well, the results are very, very evident." [continued here]
-- A high school teacher.
"Speaking as an
aerospace engineer, this was a great course, great professor [at
a different school], and great textbook. I would have never
found that textbook on my
own. I would not change anything with this course. It meets the
demands of the Math Analysis course in a previous era... the one
I grew up in. For the
most part, Calculus and Pre-Calculus classes have been
dramatically watered down since the 1980s. This course restores
the ancient paths.
"I think this is the best math
class she [his daughter] has had for her mathematical
understanding [continued here]"
-- A parent
Of course, the presentation of most topics resembles that of
other precalculus texts. So does the organization, at least
after Chapter
1 (which is unique). But it is particularly effective because of
its numerous distinguishing features:
An effective new approach that incorporates
graphing
calculators
as
a learning tool (not just a calculating tool).
This may be the first text to fully adjust to the fact that
calculators can do the calculations and therefore accelerate
learning about algebra, exponentials, logs, and trig, but students
must still learn and understand the math. Calculators
can help:
Concentrate attention on essential points
Increase the rate at which students
gather experience with the subject
Focus on learning math that is valuable (essential!) even
though calculators can do all the calculations. There
is still a lot to learn about math, even though calculators
can do a lot. This text clearly distinguishes between learning
about calculators and learning about math with the help of
calculators. (Dr. Esty has given numerous conference talks
about learning with the help of calculators.)
Emphasis on learning how thesymbolic
languageof math
is used. This is probably the most
distinguishing feature of the text. It is an explicit goal of
Chapter 1 that students learn how thoughts about methods are
written in modern mathematical notation. The goal of
Chapter 1 is to have students become able to learn math by
reading math. Let's face it, most students do not learn
math by reading it. This text has explicit reading lessons!
They will
Increase students' ability to generalize
properly
Increase students' ability to learn
outside of class. [There is so much more time outside of
class than in class. Wouldn't it be great if students couldlearn
(not just practice) outside class!]
Illuminating
homework in addition to the usual type of calculation
problems for practice.
(For example, most "B1" problems ask for an
illustration, or explanation, rather than a computation.)
Memorable
visual illustrations that generate correct concepts.
Emphasis on connections to lower- and higher-level
material.
(Calculus-style applications of algebra are
frequently discussed).
Two
sections
devoted to how to do word problems. Students who
can't do word problems are missing something important about
algebra -- how symbolism is used to represent operations. The
emphasis on symbolism for expressing thoughts about operations
in Chapter 1 helps students learn how to do word problems, and
they would never be ready without it. The author's research on
word problems shows that students cannot do word problems just
by taking years of algebra. They need to study writing about
operations in symbols first.
Emphasis on graphs and their interpretation, and
effective use of graphing calculators.
Emphasis on mathematical concepts that will not
become obsolete when the next generation of calculators
arrives.
Instructor-friendly and student-friendly
(this text does not require new teaching techniques or
classroom experiments)
Excellent for
teaching yourself. It is hard to learn math on your own.
You must have good (English) reading skills. In contrast to
all other texts, this one has lessons in Chapter 1 on how math
is written and how to read it. This should help you get the
most of the text even if you don't have a teacher.
A solution manual
with solutions to odd-numbered problems so you can follow how
problems are done if you have questions.
Six articles by Dr. Esty on learning precalculus with the
aid of calculators have appeared in the recent proceedings of
the International Conference on Technology in Collegiate
Mathematics (some are on-line with links below).
The importance of conceptual development that is
specifically algebraic is discussed in
"Algebraic Thinking,
Language, and Word Problems," an article by Dr. Esty and
Dr. Anne Teppo in the 1996 Yearbook: Communication in
Mathematics, published by the National Council of Teachers
of Mathematics. They have also written related articles on
problem-solving and algebraic thinking in several issues of Psychology
in Mathematics Education. |
e-books in this category
Theoretic Arithmetic
by Thomas Taylor, A. J. Valpy , 1816 The substance of all that has been written on this subject by Nicomachus, Iamblichus, and Boetius, together with some particulars respecting perfect, amicable, and other numbers, which are not to be found in the writings of modern mathematicians. (2212 views)
The Theory of Numbers
by R. D. Carmichael - John Wiley & Sons , 1914 The purpose of this book is to give the reader a convenient introduction to the theory of numbers. The treatment throughout is made as brief as is possible consistent with clearness and is confined entirely to fundamental matters. (5267 views)
Elementary Number Theory
by William Edwin Clark - University of South Florida , 2002 One might think that of all areas of mathematics arithmetic should be the simplest, but it is a surprisingly deep subject. It is assumed that students have some familiarity with set theory, calculus, and a certain amount of mathematical maturity. (5854 views)
Elementary Number Theory
by William Stein - Springer , 2004 Textbook on number theory and elliptic curves. It discusses primes, factorization, continued fractions, quadratic forms, computation, elliptic curves, their applications to algorithmic problems, and connections with problems in number theory. (7162 views)
An Introduction to the Theory of Numbers
by Leo Moser - The Trillia Group , 2007 The book on elementary number theory: compositions and partitions, arithmetic functions, distribution of primes, irrational numbers, congruences, Diophantine equations; combinatorial number theory, and geometry of numbers. (8377 views) |
The Maths Faculty
The Maths Faculty aims to ensure that every student maximises their potential in gaining the maths skills they need for their future.
Maths courses are changing to emphasise the practical application of maths to real life situations, as employers see this as a vital skill. A strong understanding of mathematics is an essential part of many college courses and careers. It is also increasingly important in the financial and technical aspects of everyday life. |
Mathematics 4 builds off of the concepts of Mathematics 3 but extends further by studying rates of change, logarithmic functions, polynomial and rational functions, and various forms of problem solving. This course includes an introduction to number theory and calculus. A graphing calculator is required for this course. |
MAA Review
[Reviewed by Mark Hunacek, on 02/17/2013]
The MAA Guide series — a subset of the Dolciani Mathematical Expositions — is rapidly becoming one of my favorite series of books. I like expository books that provide a quick and interesting entrée into an area of mathematics, or a useful source of examples, and that is precisely what these are. They are also, thanks to careful selection of authors, generally very well-written, informative and particularly useful as a resource for a varied audience. This book, the most recent one in the series (number 8, following books on complex variables, advanced real analysis, real variables, topology, elementary number theory, advanced linear algebra and plane algebraic curves) continues this tradition.
Each volume in this series is addressed to readers who, although mathematically sophisticated, are not experts in the subject matter of the book. The canonical example, I would think, would be graduate students seeking an efficient way of helping prepare for qualifying exams. However, faculty members who haven't had occasion to work extensively in a given area and who want a quick overview of the basic ideas and how they hang together would also find these books valuable. The emphasis in most of the Guides that I have read (this one most definitely included) is providing a survey of the subject in a reasonably short amount of pages, providing a book that is accessible and informative but likely does not contain the kind of technical detail that, although obviously necessary for complete mastery of the material, may serve as an impediment to a person who just wants to know "what's what" in an area.
So, for example, this book, like many in the Guide series (one possible exception is Weintraub's Advanced Linear Algebra) is not really intended as a text. There are no exercises, and most proofs are omitted; some that are fairly easy are provided, though never in the rigid theorem/proof format of most textbooks. Instead of proofs, Gouvêa provides discussions of the results and, quite often, a helpful sort of intuition as to why something should be true. (The author uses the phrase "shadows of proofs" in this connection.) To compensate for the lack of proofs, there is an excellent bibliography, to which the author makes frequent specific references throughout the text.
There are also lots of nice examples. A professional algebraist may be able to immediately give an example of a projective module that is not free, or a ring that does not have the invariance of basis number property, but people who don't work with algebra all the time may not have such examples on the tip of their tongues. The reader will find such examples here (along with, in connection with the latter, a succinct explanation of why such an example must be noncommutative). The reader will also find some examples that involve completely different branches of mathematics; there is, for example, a nice little one-page discussion of how modular forms arise from group actions, and the author also makes occasional remarks about topics such as topology and elliptic curves. The discussions here are not deep or technical, just brief overviews that give the reader some idea of what the terms mean; perfect for a student or non-specialist faculty member who may wind up hearing the phrase in a talk somewhere. In conformity with the intended readership, examples are not necessarily set off with big margins and the word EXAMPLE in large letters, but are often incorporated directly into the text.
The book is divided into six chapters, the first three of which are largely prefatory to the last three, which in turn comprise the meat of the book. Chapter 1 provides a succinct, interesting historical look at algebra, in which the author briefly tracks the development of algebra from its classical origins through its modern period (i.e., the axiomatic approach of Artin and Noether) up to its "ultramodern" period of category theory. Chapter 2 continues the study of categories; not being a huge fan of what Serge Lang once famously referred to as "abstract nonsense", I feared, when I saw this early chapter on the subject, that the entire book would be filled with commutative diagrams and exact sequences, but was pleased to discover, as I read on, that Gouvêa does not overdo this; these things generally don't appear unless their appearance really does enhance the discussion. Chapter 3 is a bestiary of algebraic terms, some of which are re-defined later and discussed in more detail.
The remaining three chapters discuss, in order, the three algebraic structures mentioned in the title of the text: groups, rings and fields (including skew fields). Chapter 4 on groups starts with the definition and then proceeds to discuss all of the general topics that one would expect to encounter in a first year graduate course, and perhaps a somewhat more: the chapter talks about Sylow theory, nilpotence and solvability, the word problem, group representation theory (in characteristic 0) and more. The discussion, even of elementary concepts, is done at a mathematically mature, but nonetheless accessible, level (for example, cosets of a subgroup H of a group G are defined as orbits under a certain group action), which I think is entirely appropriate, given the intended readership, and which also has the advantage of letting the reader see how these ideas really fit into the "big picture" (for example, the fact that distinct cosets partition the group is now seen to be just a special case of the more general result about orbits).
The next chapter is on rings and modules, and here, too, we are treated to an excellent survey of that area of mathematics: basic definitions, followed by discussions of topics such as localization, Weddeburn-Artin theory, the Jacobson radical, factorization theory, Dedekind domains (with a look at algebraic numbers), and various kinds of modules (free, projective, injective, etc.). As in the earlier chapter on group theory, the discussion here is at a mature level, with the author frequently stating things at a somewhat greater level of generality than might usually be encountered. (Examples: a quite general statement of Nakayama's lemma is given, and the usual results about modules over PIDs are deduced as a special case of the more general situation of modules over Dedekind domains.) Notwithstanding this, however, Gouvêa also keeps the needs of students firmly in mind; for example, there is a section titled "Traps", in which he points out, with simple specific examples, some of the ways in which modules can differ from vector spaces. (He tells of a friend who once described modules as "vector spaces with traps".)
The final chapter is on field theory. Galois theory is covered, of course (in a considerably general way, including infinite Galois groups and their topologies) but the chapter also contains material on such topics as algebras over a field, function fields, central simple algebras and the Brauer group.
Because the author is writing for people who already have some mathematical sophistication, including some prior exposure to abstract algebra, he does not feel obliged to follow a strictly linear order of presentation. So, for example, the chapter on groups, which precedes the chapters on rings and fields, nonetheless contains references to things like finite fields, semisimple rings and algebraic numbers; as another example, Nakayama's Lemma in ring theory is stated in a form involving tensor products, which are not formally discussed until a few sections later. This provides a certain freedom that an author of a strictly introductory text does not have, and helps, I think, enhance one's overall understanding of the subject by providing a broader point of view than might otherwise be possible. Likewise, even within a chapter, the level of difficulty is not necessarily monotonically increasing, and sometimes fairly sophisticated topics (e.g., profinite groups) are discussed before much more elementary ones (e.g., permutation groups). So, if you find a certain section to be fairly heavy going, just keep reading, and chances are, within a page or two, you will find things more comfortable.
The writing style throughout the book is of uniformly high quality. The author is one of those rare people who has the ability to write like people talk, with a nice, conversational tone that sometimes elicits a smile as well as a nod of understanding. Here, for example, is how he ends his discussion of groups of small order: "The next interesting case is order 16, which is, alas, a bit too interesting. There are five different abelian groups (easy to describe) and there are nine different nonabelian ones (most of them not easy to describe). So we will stop here." And see also page 160 for a cute little comment that will appeal to fans (of a certain age) that remember Tom Lehrer.
It should be apparent from the preceding discussion that I liked this book — a lot. Nevertheless, it seems inevitable that any reviewer will find some nits to pick, just because no two people will ever write the same book. The ones I have, though, are neither numerous nor particularly significant, and basically just reflect my personal preferences. I would have liked, for example, to have seen an example of non-isomorphic groups with the same character table (Everybody's Favorite Example is D4 and the quaternion group), as well as a specific example of a rational polynomial of degree 5 that is not solvable (the author states that the "generic" polynomial of degree at least five is not solvable and also states that an irreducible polynomial of prime degree with two real roots and at least one non-real root is not solvable by radicals, but does not give an actual fifth-degree polynomial meeting these conditions). I think the phrase "special linear group" should have been introduced when the group SL(n,K) was first defined on page 33, rather than fifty pages later, and also think that discussing unique factorization without at least mentioning Fermat's Last Theorem can only be described as a lost opportunity.
Additionally, one of my favorite cute applications of transcendence bases has always been the proof that the field of complex numbers has infinitely many automorphisms (a fact that I think is insufficiently well known); the author develops all the machinery necessary to establish this, but doesn't say so explicitly. Finally, in connection with the definition of algebraically closed fields, the author states the Fundamental Theorem of Algebra (that the field of complex numbers is algebraically closed) and says that all proofs "depend on the topology of the complex field". This statement, though true, may lead students to believe that all proofs are very analytic or topological in nature; in fact, there is at least one proof that uses Sylow and Galois theory and only two simple facts from analysis, namely (a) that any real polynomial of odd degree has at least one real root, and (b) that any quadratic polynomial with complex coefficients has a complex root.
But these are quibbles. Overall, this is a valuable book — a pleasure to read, and packed with interesting results. It should be very helpful to graduate students and non-specialists wanting a succinct summary of the subject, and even professional algebraists may find something new and interesting here. It is a splendid addition to an excellent series.
One final comment: in the interest of full disclosure, I should mention that, as faithful readers of this column probably already know, the author of this book is also the editor of this column. This raises, I suppose, at least the question of a conflict of interest. This same issue arose when another of the author's books, p-adic Numbers, was favorably reviewed in this column by Darren Glass more than two years ago, and since I don't think that I can improve on the way Professor Glass addressed it, I will simply quote him verbatim: "[T]he reader can rest assured that this reviewer would have said equally flattering things about the book even if it wasn't written by his editor. Besides, I couldn't think of anything that an editor could use to bribe his volunteer reviewers with (More prominent placing on the site? First crack at the new Keith Devlin?) so I didn't even bother asking." |
Basic Topology
9780387908397
ISBN:
0387908390
Pub Date: 1983 Publisher: Springer Verlag
Summary: In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of ...various difficulties will help students gain a rounded understanding of the subject |
56 minute basic algebra lesson is for the beginning algebra student or for anyone who has not recently studied algebra. It includes the language and symbols of algebra, (plus or minus ±, equal to =, not equal to ≠, approximately equal to, less than <, less than or equal to ≤, greater than >, greater than or equal to ≥) and introduces the polynomial. In this lesson you will be introduced to the variable "x", learn what a term, factor, exponent and degree of a term mean and be able to:
- understand what a polynomial, binomial, trinomial, are
- evaluate a polynomial with integers (numbers)
- simplify polynomials by collecting like terms
- simplify polynomials with brackets
- simplify polynomials using the distributive property
- do application problems such as by how much does 2x^2 -3x + 5 exceed 3 x^2 - 5x + 6
This lesson contains explanations of the concepts and 27
Recent Reviews
All the terms and concepts of Algebra are here. Was an excellent refresher for me and would probably have made my earlier math classes go by much smoother. No frills and no fluff but it gets the job done.
All the terms and concepts of Algebra are here. Was an excellent refresher for me and would probably have made my earlier math classes go by much smoother. No frills and no fluff but it gets the job done. |
Why Minus Times Minus Is Plus : The Very Basic Mathematics of Real and Complex Numbers
[Back cover text:]MATHEMATICS / ALGEBRAThis book is written for a very broad audience. There are no particular prerequisites for reading this book. We hope students of High Schools, Colleges, and Universities, as well as hobby mathematicians, will like and benefit from this book. The book is rigorous and self-contained.
All results are proved (or the proofs are optional exercises) and stated as theorems. Important points are covered by examples and optional exercises. Additionally there are also two sections called "More optional exercises (with answers)."Modern technology uses complex numbers for just about everything. Actually, there is no way one can formulate quantum mechanics without resorting to complex numbers.Leonard Euler (1707-1786) considered it natural to introduce students to complex numbers much earlier than we do today. Even in his elementary algebra textbook he uses complex numbers throughout the book.Nils K. Oeijord is a science writer and a former assistant professor of mathematics at Tromsoe College, Norway. He is the author of The Very Basics of Tensors, and several other books in English and Norwegian. Nils K. Oeijord is the discoverer of the general geneticcatastrophe (GGC).
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Edition:
2010
Publisher:
iUniverse, Incorporated
Binding:
Trade Paper
Pages:
136
Size:
7.50" wide x 9 Minus Times Minus Is Plus : The Very Basic Mathematics of Real and Complex Numbers - 9781450240635 at TextbooksRus.com. |
Basic Mathematical Tools for Imaging and Visualization
Administrative Info
(For CSE students, this lecture is credited with
2 ECTS with the restriction
that it has to be taken together with either the
Computer Aided Medical Procedures lecture or the
3D Computer Vision lecture.)
The lecture is given in English.
Time & Location
Monday, 14:00 - 15:30 Thursday, 16:00 - 17:30
The lessons will take place in room MI 03.013.010.
The programming exercises will take place in room MI 03.013.008.
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New regulation for CSE students: For CSE students, the lecture is credited with 2 ECTS with the restriction that it has to be taken together with either the Computer Aided Medical Procedures (CAMP) lecture or the 3D Computer Vision (3DCV) lecture.
Overview
In order to solve real-world problems in applied engineering areas of computer science, knowledge of basic mathematical tools is essential.
The aim of this lecture is to provide a basic mathematical toolbox for selected topics of Imaging and Visualization.
We
The lecture will have three main parts: Basics, Tools and Practise.
In the first part, we will give a reminder of linear algebra, analysis, geometry, probability and statistics basics. We go on by presenting the use of these basic concepts in methods such as parameter estimation and optimization.
And finally, the students will have the opportunity to gain a deep understanding and hands-on experience of the methods by implementing them and/or using them to solve real-world problems during the exercises.
In order to solve real-world problems in applied engineering areas of computer science, knowledge of basic mathematical tools is essential. The aim of this lecture is to provide a basic mathematical toolbox for selected topics of Imaging and Visualization. We The lecture will have three main parts: Basics, Tools and Practise. In the first part, we will give a reminder of linear algebra, analysis, geometry, probability and statistics basics. We go on by presenting the use of these basic concepts in methods such as parameter estimation and optimization. And finally, the students will have the opportunity to gain a deep understanding and hands-on experience of the methods by implementing them and/or using them to solve real-world problems during the exercises. |
books.google.com - An experienced math instructor and teacher trainer helps to make PreCalculus easy—even for students who feel intimidated by more advanced math topics. His orderly, step-by-step approach begins with concepts and skills typically introduced in a first-year high-school-level algebra course then progresses... The Easy Way |
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Geometry: A Comprehensive Course by Dan Pedoe Introduction to vector algebra in the plane; circles and coaxial systems; mappings of the Euclidean plane; similitudes, isometries, Moebius transformations, much more. Includes over 500 exercises.
The Divine Proportion by H. E. Huntley Discussion ranges from theories of biological growth to intervals and tones in music, Pythagorean numerology, conic sections, Pascal's triangle, the Fibonnacci series, and much more. Excellent bridge between science and art. Features 58 figures.
The Beauty of Geometry: Twelve Essays by H. S. M. Coxeter Absorbing essays demonstrate the charms of mathematics. Stimulating and thought-provoking treatment of geometry's crucial role in a wide range of mathematical applications, for students and mathematiciansA Manual of Greek Mathematics by Sir Thomas L. Heath This concise but thorough history encompasses the enduring contributions of the ancient Greek mathematicians whose works form the basis of most modern mathematics. Discusses Pythagorean arithmetic, Plato, Euclid, more. 1931 edition.
The Thirteen Books of the Elements, Vol. 1 by Euclid Volume 1 of 3-volume set containing complete English text of all 13 books of the Elements plus critical analysis of each definition, postulate, and proposition. Vol. 1 includes Introduction, Books I and II: Triangles, rectangles. .
The Thirteen Books of the Elements, Vol. 2 by Euclid Volume 2 of 3-volume set containing complete English text of all 13 books of the Elements plus critical analysis of each definition, postulate and proposition. Vol. 2 includes Books 3-9: Circles, relationships, rectilineal figures.
The Thirteen Books of the Elements, Vol. 3 by Euclid, Thomas L. Heath Volume 1 of 3-volume set containing complete English text of all 13 books of the Elements plus critical analysis. Covers textual and linguistic matters, more. Includes 2,500 years of commentary. Total in set: 995 figures. Geometry of Art and Life by Matila Ghyka This classic study probes the geometric interrelationships between art and life in dissertations by Plato, Pythagoras, and Archimedes and examples of modern architecture and art. 80 plates and 64 figures.
A Mathematical History of the Golden Number by Roger Herz-Fischler This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.
Greek Mathematical Thought and the Origin of Algebra by Jacob Klein Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th-16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. 1968 edition. Bibliography.
From Geometry to Topology by H. Graham Flegg Introductory text for first-year math students uses intuitive approach, bridges the gap from familiar concepts of geometry to topology. Exercises and Problems. Includes 101 black-and-white illustrations. 1974A Treatise on Algebraic Plane Curves by Julian Lowell Coolidge A detailed introduction to the theory of algebraic plane curves and their relations to various fields of geometry and analysis, this text employs both algebraic and geometric methods. 1931 edition. 17 illustrations.
Product Description:
accurate depictions of large or small conic curves and describing their reflective properties, from light in telescopes to sound in microphones and amplifiers. It further defines the role of curves in the construction of auditoriums, antennas, lamps, and numerous other design applications. Only a basic knowledge of plane geometry needed; suitable for undergraduate courses. 1993 edition. 98 |
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How
to Use the Site?
The contents of the site embrace the whole
mathematics curriculum of the high school. You'll find here
theory, problems and tests. All the theoretical questions are
subdivided into groups with sets of problems for independent
solution and repetition of the studied material. The formulation
of every problem is provided with the links, which lets you
get help, see the solution and compare with the answer. The
contents of the site are constantly being refilled.
On the left side of the page you can see the main school sections
of the site.
"Search Site"helps you to find the page of the site, which
contains a wanted theme.The rules of using of the search
engine are written in details on the page Search .
"Program of Lessons" provides to study the mathematics curriculum by list of lessons . Click for this the button Lessons.
"Problems" ( the button Select
topic ) provides you transit to the menu, which contains a list of problems
of all the themes. Click any position and you receive a set of problems of different
difficulty you're interested in. If it's necessary you can click a suitable link
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You can check if you're prepared entering the section "Tests & Exams".
Click the button Select
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This chapter is about arithmetic.
Arithmetic traditionally includes procedures for integer
addition, subtraction, multiplication, and division.
Here we will discuss procedures for interval operations.
The discussion will not be of particular operations,
such as addition or multiplication. Rather, properties of
common operations, such as local concavity, will drive
the discussion. This will lead toward a framework for
implementing generalized interval models of real functions. |
Mathematics Courses:
Catalog Description:
Students move through a series of content modules using a mastery learning approach, making extensive use of computer software for content delivery, practice of skills, and assessment (graded homework, quizzes, and module pre- and post-tests). Topics include operations with whole numbers, fractions, decimals, and signed numbers; ratios, rates, and proportions; percent; measurement conversions within the U. S. Customary System and within the Metric System using Unit Analysis; scientific notation; evaluating and simplifying variable expressions; solving linear equations; and some basic geometry. Prerequisite: By placement
Lecture: 3 hrs.
Course Student Learning Outcomes (CSLOs):
Upon successful completion of the course, the student will be able to demonstrate each of the following skills in writing on one or more tests without referring to notes, textbook, or other resources:
* 1. Use the Order of Operations Agreement to correctly evaluate numerical expressions involving integers and/or fractions and any combination of addition, subtraction, multiplication, division, whole number exponents, and parentheses.
2. Write the prime factorization of a given natural number greater than 1 and less than 1000.
* 3. Correctly evaluate expressions involving addition, subtraction, multiplication, and division of pairs of numbers from the following categories and express the answers in simplest form: a) integers, b) decimal numbers, c) fractions with unlike denominators and mixed numbers (in combination with each other and with whole numbers).
4. Use the symbols < , > , and = to correctly identify the size relation between pairs of integers, pairs of fractions with unlike denominators, pairs of decimals, and a fraction compared with a decimal.
5. Correctly round a given whole number or decimal number to a given place value.
6. a) Using words, correctly write the name of a give whole number, less than one quadrillion, that is shown written in standard decimal form.
b) Using words, correctly write the name of a given positive number, less than one, which is shown written in standard decimal form.
c) Correctly write a whole number, less than one quadrillion, in standard decimal form when given its name written in words.
d) Correctly write a positive number, less than one, in standard decimal form when given its name written in words.
7. a) When given two quantities (measured with either the same or different units), correctly express their relationship using a ratio or rate written in simplest fraction form.
b) Correctly write a unit rate involving two given quantities.
8. Correctly set up and solve a proportion when given a verbal description of a proportional situation in which one quantity is unknown.
9. Correctly convert a given number from one of the following forms to each of the other two forms:
a) Percent, b) fraction/mixed number, c) decimal.
10. Find the missing percent, base, or amount when given the other two using either the percent equation
percent . base = amount or the percent proportion .
12. a) Correctly convert given numbers greater than 10 from scientific notation to standard decimal form.
b) Correctly convert given numbers less than 0.1 from scientific notation to standard decimal form.
c) Correctly convert given numbers greater than 10 from standard decimal form to scientific notation.
d) Correctly convert given numbers less than 0.1 from standard decimal form to scientific notation.
13. Correctly use the Distributive Property, Commutative and Associative Properties of Addition and Multiplication, and combining like terms to simplify a variable expression.
14. Correctly solve linear equations in one variable that may include variable and/or constant terms on both sides of the equation and may include expressions in parentheses.
15. Write a variable expression or equation that correctly represents an expression described verbally in writing.
16. a) Convert measurements of length/distance, weight, and capacity to equivalent measurements with different sized units within the U.S. Customary System.
b) Convert measurements of length/distance, mass, and capacity to equivalent measurements with different sized units within the Metric System (S.I.).
* This course objective has been identified as a student learning outcome that must be formally assessed as part of the Comprehensive Assessment Plan of the college. All faculty teaching this course must collect the required data and submit the required analysis and documentation at the conclusion of the semester to the Office of Institutional Research and Assessment.
IV. Measurement Conversions
Conversions of units of measurement of length/distance, weight, capacity, and time within the U. S. Customary System using Unit Analysis; conversions of units of measurement of length/distance, mass, and capacity, within the Metric System (S. I.)
VI. Percent
Converting between percent form and fraction form; converting between percent form and decimal form; solving percent problems using either the percent equation
percent . base = amount or the percent proportion |
reflects the new excitement in Elementary Differential Equations as the availability of specialized scientific computing environments and software systems continues to reshape the role and applications of the discipline. |
Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way through a course. Sometimes it feels as though I've walked into class and said "Forget math. Let's learn ancient Greek instead." Sometimes the students realize that Greek is interesting too, but it can take a lot of convincing! Hence I would really like to let students know, right from the start, what they're getting themselves into.
Does anyone know of a text that might help me do this in a not-too-advanced manner? One possibility, I guess, is Linear Algebra Done Right by Axler, but are there others? Axler's book might be too advanced.
Or would anyone caution me against trying this, based on past experience? |
hostie
05-15-2009, 10:50 PM
Hmm, there really isn't a lot of Pre-Calc out there... Basic math up to grade 11?
Why don't you just find a calculus book and learn some of it on your own? The first part of it really isn't a big deal..
The Jitterskull
05-15-2009, 10:53 PM
How far are you looking for? Into integration? Derivatives? Trig?
skyclaw441
05-15-2009, 11:35 PM
Just to get me through first semester, so I can explore it more on my own later on as well.
Zip
05-16-2009, 07:05 AM
Anal Assassin
05-19-2009, 04:41 AM
first semester is mostly detailed review of concepts you learned in algebra two. After which is trigonometry.
I can help you with trig, I think I have pretty good notes that lay out the foundations pretty nicely, starting with the unit circle and onto all 36 identities.
Looking through my first semester notes now here is some of the general overview.
2nd semester is mostly trigonometry.
1.unit circle
2. radians
3. converting between them and degrees
4. sine cosines tangent
5. secants, cosecants and cotangents
6. graphing all 6
7. proving equality
8. and you will end the semester off in vectors
skyclaw441
05-19-2009, 04:43 AM
Ahh! Fuck! I hate Algebra II. But, I guess I could just pay extra attention to what's on the final. I'm good at the basic trig I did in Geometry, so trig doesn't sound terrible.
el drewto
05-23-2009, 04:20 AM
You have a graphing calculator, right?
reallystupidstuff
06-01-2009, 12:15 AM
I believe a pre-calculator was called an abacus
skyclaw441
06-01-2009, 04:04 AMIf you want something to use in class, I'd recommend at least a TI-86, which should take you through basic calculus. If you're not looking for something to use in class (as others have said), just use the Internet. |
Algebra I
The first year of algebra is a prerequisite for all higher-level math: geometry, algebra II, trigonometry, and calculus. Studies have argued that students who take algebra I, geometry, algebra II, and one additional high-level math course are much more likely to do well in college math, than students without.
According to GreatSchools.org Algebra is not just for the college-bound. Even high school graduates headed straight for the work force need the same math skills as college freshmen, the ACT found. One particular study looked at occupations that don't require a college degree but pay wages high enough to support a family of four. Researchers found that math and reading skills required to work as an electrician, plumber, or upholsterer were comparable to those needed to succeed in college. In summery a strong algebra background is a must for our 21st century.
Basic algebra is the first in a series of higher-level math classes students need to succeed in college and life. In order to build a solid math foundation, there are several strong resources we would encourage you to take a look at. Each subtopic has links to useful discussions on the TFANet content community (Yenche Tioanda is a godsend) and several files to help you design successful lessons. |
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