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Basic Algebra (SSAT-Algebra)
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. |
Vaughn, WA AlgebraMath and Science are the means we use to communicate the nature of our world to one another. Having a better understanding of math gives us a better understanding of our natural world. That part of the world that we can measure |
Username: this is the same as your LHUP computer username.
Institution: lhup
Password: this will be the same as your username. You can, and should, change this once you log on.
Obtain a WebAssign Access Code You will have to purchase an Access Code Online
Notes on entering information using WebAssign
1. scientific notation - WebAssign uses scientific notation like that used in calculators. Some examples:
number
scientific notation
WebAssign Notation
12234
1.2234x104
1.2234E+4
0.00344
3.44x10-3
3.444E-3
Exponents - You may be asked to enter variabls with exponents. For example, L2. The way
to enter this into your answer is as follows:
L^2
3. precision -Unless the problems states otherwise, WebAssign will allow a 2% tolorance in your answers.
4. angles - WebAssign measures angles from the positive x-axis.
Going counterclockwise from the +x-axis use values from 0 to +1800 (positive values).
Going clockwise from the +x-axis use values from 0 to -1800 (negative values).
The figure below shows an example. There area 4 vectors. The angles of these vectors, according to WebAssign,
would be referenced to the +x axis. WebAssign would say the vectors are located at the following angles:
WebAssign
Vector A is located at angle +30o from the +x axis
Vector B is located at angle +150o from the +x axis
Vector C is located at angle -150o from the +x axis
Vector D is located at angle -30o from the +x axis |
Algebra 2 is a college-preparatory class that focuses on the study of functions - linear, quadratic, polynomial, rational, exponential and logarithmic. A graphing calculator is required to fully explore the concepts of this course. |
AcademicsCombinatorics
Combinatorics is the study of arranging
objects to satisfy given rules. That is, we try to solve puzzles
such as:
a) Take a chessboard and remove a pair of
diagonally opposite corner squares. Suppose we have some dominos,
each of which exactly covers two adjacent squares of the chessboard.
Is it possible to exactly cover this chessboard using the
dominos?
b) Kirkman's Schoolgirl Problem (1850):
A schoolmistress takes her class of 15 girls on a daily walk.
The girls are arranged in 5 rows, with 3 girls in each row,
so that each girl has 2 companions. Is it possible to plan
a walk for 7 consecutive days so that no girl will walk in
a triplet with any of her classmates more than once?
c) Suppose a region of land is to be divided
into countries. How many colors do we need so that, however
the region is divided, we can produce a map of the region
in which no two neighboring countries are the same color?
If we can show that a certain arrangement
exists, then we are interested in how many possible arrangements
there are. In the course we study some essential techniques
of combinatorics and then look at some problems from various
areas of the field. We also consider how combinatorial techniques
and results apply to other fields, such as experimental design
and computer science.
Course Structure: The course text
is "Introductory Combinatorics" (3rd edition) by
Richard Brualdi (available in bookstore and on reserve in
the library). An approximate outline for the course
is chapters 1-3, 5, 6, 9-12 of the text. There
will be two types of homework assignment. The first
type will consist of problems which reinforce and build on
the skills we are developing in class; the second type will
be a single challenging problem. They will be set on
alternate Thursdays. The first type are due by 4pm the
following Wednesday. For the second type there is nothing
to hand in, but you should come to class the following Thursday
ready to discuss progress you have made and approaches you
have attempted.
Grading: Your grade will be based on 3 equally
weighted components; the first type of homework, a final exam
and class participation. The class participation component
will rely heavily on your engagement with the problems in
the second type of homework. |
Subject overview
Why mathematics?
Mathematics is core to most modern-day science, technology and business. When you turn on a computer or use a mobile phone, you are using sophisticated technology that mathematics has played a fundamental role in developing. Unravelling the human genome or modelling the financial markets relies on mathematics.
As well as playing a major role in the physical and life sciences, and in such disciplines as economics and psychology, mathematics has its own attraction and beauty. Mathematics is flourishing: more research has been published in the last 20 years than in the previous 200, and celebrated mathematical problems that had defeated strenuous attempts to settle them have recently been solved.
The breadth and relevance of mathematics leads to a wide choice of potential careers. Employers value the numeracy, clarity of thought and capacity for logical argument that the study of mathematics develops, so a degree in mathematics will give you great flexibility in career choice.
Why mathematics at Sussex?
Mathematics at Sussex was ranked in the top 20 in the UK in The Sunday Times University Guide 2012.
In the 2008 Research Assessment Exercise (RAE) 90 per cent of our mathematics research and 97 per cent of our mathematics publications were rated as recognised internationally or higher, and 50 per cent of our research and 64 per cent of our publications were rated as internationally excellent or higher.
The Department awards prizes for the best student results each year, including £1,000 for the best final-year student.
In 2011, US careers website Jobs ratedranked mathematician to be the second most popular job out of the 200 considered.
You will find that our Department is a warm, supportive and enjoyable place to study, with staff who have a genuine concern for their students.
Our teaching is informed by current research and understanding and we update our courses to reflect the latest developments in the field of mathematics.
MMath or BSc?
The MMath courses are aimed at students who have a strong interest in pursuing a deeper study of mathematics and who wish to use it extensively in careers where advanced mathematical skills are important, such as mathematical modelling in finance or industry, advanced-level teaching or postgraduate research.
Applicants unsure about whether to do an MMath or a BSc are strongly advised to opt initially for the MMath course. If your eventual A level grades meet the offer level for a BSc but not an MMath we will automatically offer you a place on the BSc course. Students on the MMath course can opt to transfer to the BSc at the end of the second year.
Why economics?
Addressing many of the world's problems and issues requires an understanding of economics. Why are some countries so rich and others so poor? Should Microsoft be broken up? Should the private sector be involved in providing health and education? Could environmental taxes help reduce global warming? What is the future of the euro?
Economics provides a framework for thinking about such issues in depth, allowing you to get to the heart of complex, topical problems. The methods of economics can be applied to a wide range of questions and will prove useful to you in your future career. In addition, the study of economics teaches you a variety of practical skills, including the ability to use and evaluate evidence (often statistical) in order to arrive at sound conclusions.
Why economics at Sussex?
In the 2008 Research Assessment Exercise (RAE) 100 per cent of our economics research was rated as recognised internationally or higher, and 60 per cent rated as internationally excellent or higher.
We emphasise the practical application of economics to the analysis of contemporary social and economic problems.
We have strong links to the major national and international economic institutions such as the European Commission, the World Bank and the Department for International Development.
The Department has strong research clusters in labour markets and in development economics, and is one of Europe's leading centres for research on issues of international trade.
We offer you the chance to conduct an economics research project supervised by a faculty member.
Programme content
This degree exploits the strong relationship between mathematical modelling and economics. Alongside the mathematics core modules, you study the principles of economic analysis and its policy applications at both the macro (economy-wide) and the micro (individual/ company) levels. The economics element provides an opportunity to acquire practical skills and to apply mathematical methods.
As well as the core mathematics modules in Years 1 and 2, you will spend 25 per cent of your time studying economics modules. In the third year, you take a combination of mathematics and economics options.
On the MMath course, you carry out a project in the fourth year and choose from a range of more advanced mathematical modules recognise that new students have a range of mathematical backgrounds and that the transition from A level to university-level study can be challenging, so we have designed our first-term modules to ease this. Although university modes of teaching place more emphasis on independent learning, you will have access to a wide range of support from tutors.
Teaching and learning is by a combination of lectures, workshops, lab sessions and independent study. All modules are supported by small-group teaching in which you can discuss topics raised in lectures. We emphasise the 'doing' of mathematics as it cannot be passively learnt. Our workshops are designed to support the solution of exercises and problems.
Most modules consist of regular lectures, supported by classes for smaller groups. You receive regular feedback on your work from your tutor. If you need further help, all tutors and lecturers have weekly office hours when you can drop in for advice, individually or in groups. Most of the lecture notes, problem sheets and background material are available on the Department's website.
Upon arrival at Sussex you will be assigned an academic advisor for the period of your study. They also operate office hours and in the first year they will see you weekly. This will help you settle in quickly and offers a great opportunity to work through any academic problemsexcellent training in problem-solving skills
understanding of the structures and techniques of mathematics, including methods of proof and logical arguments
written and oral communication skills
organisational and time-management skills
an ability to make effective use of information and to evaluate numerical data
IT skills and computer literacy through computational and mathematical projects
you will learn to manage your personal professional career development in preparation for further study, or the world of work.
Core content
Year 1
You take modules on topics such as calculus • introduction to pure mathematics • geometry • analysis • mathematical modelling • linear algebra • numerical analysis. You also work on a project on mathematics in everyday life.
Year 2
You take modules on topics such as calculus of several variables • an introduction to probability • further analysis • group theory • probability and statistics • differential equations • complex analysis • further numerical analysis core ideas and analytical techniques are presented in lectures and supplemented by classes or workshops where you can test your understanding and explore the issues in more depth. These provide the opportunity for student interaction, an essential part of the learning process at Sussex. The more quantitative skills, such as using statistical software, are taught in computer workshops. On the dissertation module in the final year, you receive one-to-one supervision as you investigate your chosen research topic in depth.
Formal assessment is by a range of methods including unseen exams and coursework. In addition there are regular assignments, which allow you to monitor your progress. In the first year, you have regular meetings with your academic advisor to discuss your academic progress and to receive feedback on your assignments and understanding of the principles of economics
the skills to abstract the essential features of a problem and use the framework of economics to analyse it
the ability to evaluate and conduct your own empirical research
the confidence to communicate economic ideas and concepts to a wider audience
a range of transferable skills, applicable to a wide variety of occupations.
Core content
Year 1
You are introduced to the principles of economics and their application to a range of practical and topical issues. The aim is not to look at economic theory in isolation but to learn how it is used to analyse real issues. You also take a mathematics module, giving you some of the tools you need to understand contemporary economics.
Year 2
You develop your understanding of economics principles through the study of more advanced topics such as trade and risk. You also take a statistics module and learn how to analyse and interpret data. In addition, there are more applied modules, allowing you to see how the subject deals with empirical issues. There are opportunities for small research projects, including a group project.
Year 3
You have the opportunity to choose from a range of options such as labour or development economics. These modules go into the relevant issues in greater depth, giving you a high level of expertise. There is the opportunity to do a sustained piece of research on a chosen topic. You can also take more advanced quantitative modules – useful if you wish to do postgraduate work.
Geometry
15 credits
Autumn teaching, Year 1
Topics include: vectors in two and three dimensions. Vector algebra: addition, scalar product, vector product, including triple products. Applications in two- and three-dimensional geometry: points, lines, planes, geometrical theorems. Area and volume. Linear dependence and determinants. Polar co-ordinates in two and three dimensions. Definitions of a group and a field. Polynomials. Complex numbers, Argand plane, De Moivre's theorem. Matrices: addition, multiplication, inverses. Transformations in R^2 and R^3: isometries. Analytical geometry: classification and properties of conics.
Introduction to Economics
15 credits
Autumn teaching, Year 1
This course provides an introduction to the fundamental principles of economics. The first half of the course deals with microeconomic issues including the behaviour of individuals and firms, their interaction in markets and the role of government. The second half of the course is devoted to macroeconomics and examines the determinants of aggregate economic variables, such as national income, inflation, and the balance of payments, and the relationships between them. This course also provides students with a basic introduction to mathematical economics, covering solving linear equations, differential calculus, and discounting.
Microeconomics 1
15 credits
Spring teaching, Year 1
This module develops consumer and producer theory, examining such topics as consumer surplus, labour supply, production and costs of the firm, alternative market structures and factor markets. It explores the application of these concepts to public policy, making use of real-world examples to illustrate the usefulness of the theory.
Numerical Analysis 1
15 credits
Spring teaching, Year 1
This module covers topics such as:
Introduction to Computing with MATLAB
Basic arithmetic and vectors, M-File Functions, For Loops, If and else, While statements
Analysis 2
15 credits
Autumn teaching, Year 2
Topics covered: power series, radius of convergence; Taylor series and Taylor's formula; applications and examples; upper and lower sums; the Riemann integral; basic properties of the Riemann integral; primitive; fundamental theorem of calculus; integration by parts and change of variable; applications and examples. Pointwise and uniform convergence of sequences and series of functions: interchange of differentiation or integration and limit for sequences and series; differentiation and integration of power series term by term; applications and examples. Metric spaces and normed linear spaces: inner products; Cauchy sequences, convergence and completeness; the Euclidean space R^n; introduction to general topology; applications and examples.
First order PDEs: Method of characteristics for semilinear and quasilinear equations, initial boundary value problems.
Macroeconomics 1
15 credits
Spring teaching, Year 2
This module introduces core short-run and medium-run macroeconomics.
First you will consider what determines demand for goods and services in the short run. You will be introduced to financial markets, and outline the links between financial markets and demand for goods. The Keynesian ISLM model encapsulates these linkages. Second, you will turn to medium-term supply. You will bring together the market for labour and the price-setting decisions of firms in order to build an understanding of how inflation and unemployment are determined. Finally, you will look at supply and the ISLM together to produce a full medium-term macroeconomic model.
Microeconomics 2
15 credits
Autumn teaching, Year 2
This module develops the economics principles learned in Microeconomics 1. Alternative market structures such as oligopoly and monopolistic competition are studied and comparisons drawn with perfect competition and monopoly. Decision-making under uncertainty and over multiple time periods is introduced, relaxing some of the restrictive assumptions made in the level 1 module. The knowledge gained is applied to such issues as investment in human capital (eg education), saving and investment decisions, insurance and criminal deterrence.
Distribution theory: Chebychev's inequality, weak law of large numbers, distribution of sums of random variables, t,\chi^2 and F distributions;
Confidence intervals;
Statistical tests including z- and t-tests, \chi^2 tests;
Linear regression;
Nonparametric methods;
Random number generation;
Introduction to stochastic processes.
Macroeconomics 2
15 credits
Autumn teaching, Year 3
This module is concerned with two main topics. 'The long run' is an introduction to how economies grow, gradually raising the standard of living, decade by decade. Once we have the basic analysis in place, we can begin to explain why there are such huge disparities in living standards around the world. 'Expectations' is a deepening of the behavioural background to modelling, saving and investment decisions, emphasising the intrinsically forward-looking nature of saving and investment decisions and analysing the financial markets which coordinate these decisions.
Advanced Macroeconomics
15 credits
Spring teaching, Year 3
The module completes the macroeconomics sequence, starting with a consideration of the policy implications of rational expectations. The macroeconomy is then opened up to international trade and capital movements: the operation of monetary and fiscal policies and the international transmission of disturbances under fixed and flexible exchange rates are contrasted, and the issues bearing on the choice of exchange-rate regime are explored. The major macroeconomic problems of hyperinflation, persistent unemployment and exchange-rate crises are examined. The module concludes by drawing together the implications of the analysis for the design and operation of macroeconomic policy.
Advanced Microeconomics
15 credits
Spring teaching, Year 3
This module covers the topics of general equilibrium and welfare economics, including the important issue of market failure. General equilibrium is illustrated using Sen's entitlement approach to famines and also international trade. Welfare economics covers concepts of efficiency and their relationship to the market mechanism. Market failure includes issues such as adverse selection and moral hazard, and applications are drawn from health insurance, environmental economics and the second-hand car market.
Harmonic Analysis and Wavelets
15 credits
Autumn teaching, Year 4
You will be introduced to the concepts of harmonic analysis and the basics of wavelet theory: you will discuss the concepts of normed linear spaces and Hilbert spaces, with a focus on sequence spaces and spaces of functions, most notably the space of square-integrable functions on an interval or on the real line. You will be introduced to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space.
You will then apply these concepts to the concrete case of classical trigonometric Fourier series, and both Fejer's theorem and the Weierstrass approximation theorem will be proved.
Finally, you will apply the introduced concepts for Hilbert to discuss wavelet analysis for the example of the Haar wavelet and the Haar scaling function. You will be introduced to the concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function for the case of the Haar wavelet), but will also be defined in general. The concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function) will initially be introduced for the case of the Haar wavelet, but will also be defined in general.
Introduction to Mathematical Biology
15 credits
Autumn teaching, Year 4
The module will introduce you to the concepts of mathematical modelling with applications to biological, ecological and medical phenomena. The main topics will include:
Perturbation theory and calculus of variations
15 credits
Spring teaching, Year 4
The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasisedRing Theory
15 credits
Autumn teaching, Year 4
In this module we will explore how to construct fields such as the complex numbers and investigate other properties and applications of rings.
Special topics: Quaternions, valuations, Hurwitz ring, the four squares theorem.
Topology and Advanced Analysis
15 credits
Spring teaching, Year 4
This module will introduce you to some of the basic concepts and properties of topological spaces. The subject of topology has a central role in all of Mathematics and having a proper understanding of its concepts and main theorem is essential as part of an undergraduate mathematics curriculum.
Topics that will be covered in this module include:
Topological spaces
Base and sub-base
Separation axioms
Continuity
Metrisability
Completeness
Compactness and Coverings
Total Boundedness
Lebesgue numbers and Epsilon-nets
Sequential Compactness
Arzela-Ascoli Theorem
Montel's theorem
Infinite Products
Box and Product Topologies
Tychonov Theorem.
MMath Project
30 credits
Autumn & spring teaching, Year 4
The work for the project and the writing of the project report will have a major role in bringing together material that you have mastered up to Year 3 and is mastering in Year 4. It will consist of a sustained investigation of a mathematical topic at Masters' level. The project report will be typeset using TeX/LaTeX (mathematical document preparation system). The use of mathematical typesetting, (mathematics-specific) information technology and databases and general research skills such as presentation of mathematical material to an audience, gathering information, usage of (electronic) scientific libraries will be taught and acquired during the project.
E-Business and E-Commerce Systems
15 credits
Autumn teaching, Year 4
This module will give you a theoretical and technical understanding of the major issues for all large-scale e-business and e-commerce systems. The theoretical component includes: alternative e-business strategies; marketing; branding; customer relationship issues; and commercial website management. The technical component covers the standard methods for large-scale data storage, data movement, transformation, and application integration, together with the fundamentals of application architecture. Examples focus on the most recent developments in e-business and e-commerce distributed systems.
Financial Portfolio Analysis
15 credits
Spring teaching, Year 4
You will study valuation, options, asset pricing models, the Black-Scholes model, Hedging and related MatLab programming. These topics form the most essential knowledge for you if you intend to start working in the financial fields. They are complex application problems. Your understanding of mathematics should be good enough to understand the modelling and reasoning skills required. The programming element of this module makes complicated computations manageable and presentable.
Harmonic Analysis and Wavelets
15 credits
Autumn teaching, Year 4
This module introduces you to the concepts of harmonic analysis and the basics of wavelet theory. We will discuss the concepts of normed linear spaces and Hilbert spaces, with a focus on sequence spaces and spaces of functions, most notably the space of square-integrable functions on an interval or on the real line. You will be intoroduces to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space, and apply these to the concrete case of classical trigonometric Fourier series. You will also use these strategies to prove both Fejer's theorem and the Weierstrass approximation theorem. Finally you will apply the concepts for Hilbert spaces to discuss wavelet analysis using the example of the Haar wavelet and the Haar scaling function. The concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function) will initially be introduced for the case of the Haar wavelet, but will also be defined in general.
Introduction to Cosmology
15 credits
Autumn teaching, Year 4
This module covers:
observational overview: In visible light and other wavebands; the cosmological principle; the expansion of the universe; particles in the universe.
cosmological models: solving equations for matter and radiation dominated expansions and for mixtures (assuming flat geometry and zero cosmological constant); variation of particle number density with scale factor; variation of scale factor with time and geometry.
inflation: definition; three problems (what they are and how they can be solved); estimation of expansion during inflation; contrasting early time and current inflationary epochs; introduction to cosmological constant problem and quintessence.
initial singularity: definition and implications.
connection to general relativity: brief introduction to Einstein equations and their relation to Friedmann equation.
Mathematical Models in Finance and Industry
15 credits
Spring teaching, Year 4
Topics include: partial differential equations (and methods for their solution) and how they arise in real-world problems in industry and finance. For example: advection/diffusion of pollutants, pricing of financial options.
Object Oriented Programming
You will be introduced to object-oriented programming, and in particular to understanding, writing, modifying, debugging and assessing the design quality of simple Java applications.
You do not need any previous programming experience to take this module, as it is suitable for absolute beginnersTechnology-Enhanced Learning Environments
15 credits
Spring teaching, Year 4
This module emphasises learner-centred approaches to the design of educational and training systems. The module content will reflect current developments in learning theory, skill development, information representation and how individuals differ in terms of learning style. The module has a practical component, which will relate theories of learning and knowledge representation to design and evaluation. You will explore the history of educational systems, as well as issues relating to: intelligent tutoring systems; computer-based training; simulation and modelling environments; programming languages for learners; virtual reality in education and training; training agents; and computer-supported collaborative learning
Specific entry requirements: A levels must include both Mathematics and Further Mathematics, grade A.
International Baccalaureate
Typical offer: 35 points overall
Specific entry requirements: Higher Levels must include Mathematics, with a grade of 6.
Advanced Diploma
Typical offer: Pass with grade A in the Diploma and A in the Additional and Specialist Learning.
Specific entry requirements: The Additional and Specialist Learning must be an A level in Mathematics (grade A). Successful applicants will also need to take A level Further Mathematics as an extra A level software development, actuarial work, financial consultancy, accountancy, business research and development, teaching, academia and the civil service. All of our courses give you a high-level qualification for further training in mathematics.
Recent graduates have taken up a wide range of posts with employers including:
actuary at MetLife
assistant accountant at World Archipelago
audit trainee at BDO LLP UK
credit underwriter at Citigroup
graduate trainee for aerospace and defence at Cobham plc
pricing analyst at RSA Insurance Group plc
assistant analytics manager at The Royal Bank of Scotland
associate tutor at the University of Sussex
health economics consultant at the University of York
risk control analyst at Total Gas & Power
supply chain manager at Unipart Group
technology analyst at J P Morgan
digital marketing consultant at DC Storm
junior financial advisor at Barclays
audit associate at Ernst & Young
claims graduate trainee at Lloyds of London
development analyst at Axa PPP healthcare
fraud analyst at American Express
futures trader at Trading Tower Group Ltd
accountant at KPMG Mathematical and Physical Sciences
The School of Mathematical and Physical Sciences brings together two outstanding and progressive departments - Mathematics, and Physics and Astronomy. It capitalises on the synergy between these subjects to deliver new and challenging opportunities for its students and faculty advice |
About Hp Calculator
Hewlett Packard offers three main types of calculators: scientific, financial, and graphing. For students in middle school, scientific calculators are the most popular, as they provide basic arithmetic functions and many more advanced computations. Graphing and financial calculators provide the additional computational power required by students in high schools and colleges, as well as professionals.In addition to arithmetic functions, all HP calculators include trigonometric, logarithmic, exponential, and power functions. These functions form the core of the math courses through which students progress, making HP calculators valuable tools. In addition, some HP calculators incorporate statistical functions such as sums, means, and standard deviations. Some HP calculators also include Reverse Polish Notation, a data entry format that reduces the number of necessary keystrokes.When scientific calculators don't provide you with the computational power that you need, graphing calculators are the next step. These provide users with a larger screen on which you can view equations and edit them. In addition to being able to graph equations in Cartesian and polar coordinates, these HP calculators have thousands of built-in functions. Even better, with the built-in memory of HP graphing calculators, users can write their own programs to solve problems and automate repeated calculations.HP financial calculators are used by both students and professionals, and include specialized financial functions. Options are immediately available to calculate loan payments, interest rates, net present value, and annuities. In addition to being used by bank officers and actuaries, these calculators are also approved for use on professional examinations such as Certified Financial Planner (CFP) and Chartered Financial Analyst (CFA) exams. |
Mathematica: Linear Algebra
Linear algebra is about the solution of simultaneous linear
equations, linear eigensystems etc. There are two courses described
here, which vary very considerably in difficulty.
Neither are currently being given and may be out of date.
Numerical Linear Algebra
This course starts by covering Mathematica's basic matrix facilities,
fairly quickly, for people who are not experts with them. It then
covers the basics of linear algebra using real and complex
matrices. It is intended for people who can use Mathematica, but need to
know what it can do with matrices and linear algebra.
Matrix arithmetic and how matrices are used in linear algebra
(e.g. the solution of linear equations) is no longer taught in the
ordinary mathematics A-level, but only Further Pure mathematics. If you
do not know this, you MUST learn it first. For further
information, see Matrix
Prerequisites.
Lectures
Symbolic Linear Algebra
The second lecture covers linear algebra with symbolic matrices, and its
debugging and tuning. It shows how to get first- and second-order
approximations to problems, with the variations in the input being in
the form of unknown variables. This enables what is often called
perturbation analysis or sensitivity analysis.
People who want to do comparable work with other numerical methods
(such as PDEs, ODEs or optimisation) will also find it relevant, as the
techniques the course covers apply to those as well.
WARNING: this sort of work is never easy, though it
is much easier in Mathematica than by hand, and attendees
should be reasonably familiar with using Mathematica. This is quite
possibly the hardest actual programming course in the University, not
excluding those given in the Computer Laboratory. They give harder
theoretical ones, of course. |
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
New Senior Mathematics for Years 11 & 12 is part of a new edition of the well-known mathematics series for New South Wales. We've completely updated the series for today's classrooms, continuing the much-loved approach to deliver mathematical rigour with challenging student questions. The first three chapters of this student book contain revision material, providing the necessary foundation for the development of senior mathematical concepts.
Table of contents
Arithmetic and surds
Algebra
Equations and inequalities
Plane geometry
Trigonometic ratios and applications
Coordinate geometry - straight lines
Functions and relations
Locus and regions
Quadratic functions and the parabola
Differential calculus
Plane and coordinate geometry
Geometrical applications of differentiation
Integral calculus
Exponential and logarithmic functions
Trigonometic functions
Series and applications
Applications of calculus to the physical world
Probability
Features & benefits
Improved grading of questions
Clearer worked examples
5 exam-style papers
Chapter reviews for each chapter
Technology tips and suggested GeoGebra use
Summary section
Series overview
A feature of NSW classrooms for almost 30 years, the New Senior Mathematics books make up the most respected and trusted senior Mathematics series in the market. |
Tibi's Mathematics Suite 2.0 Beta Downloading Page
Tibi's Mathematics Suite was developed as a simple, accessible and very easy-to-use piece of software that allows all the users to perform various mathematical calculations
Tibi's Mathematics Suite 2.0 Beta External Mirrors
Tibi's Mathematics Suite 2.0 Beta is a Freeware. Please read this article and discover
what exactly does Freeware mean.
Whether you're happy or not testing and using Tibi's Mathematics Suite 2.0 Beta, be our guest and let's solve all the problems related to this software together. Feel free to use:
Tibi's Mathematics Suite 2 |
Syllabus Structure and Content
3.1 INTRODUCTION
The way in which the mathematical content in the syllabuses is organised and presented in the syllabus document is described in sections 3.2 and 3.3. Section 3.3 also discusses the main alterations, both in content and in emphasis, with respect to the preceding versions. The forthcoming changes in the primary curriculum, which will have "knock-on" effects at second level, are outlined in Section 3.4. Finally, in Section 3.5, the content is related to the aims of the syllabuses.
3.2 STRUCTURE
For the Higher and Ordinary level syllabuses, the mathematical material forming the content is divided into eight sections, as follows:
Sets
Number systems
Applied arithmetic and measure
Algebra
Statistics
Geometry
Trigonometry
Functions and graphs
The corresponding material for the Foundation level syllabus is divided into seven sections; there are minor differences in the sequence and headings, resulting in the following list:
Sets
Number systems
Applied arithmetic and measure
Statistics and data handling
Algebra
Relations, functions and graphs
Geometry
The listing by content area is intended to give mathematical coherence to the syllabuses, and to help teachers locate specific topics (or check that topics are not listed). The content areas are reasonably distinct, indicating topics with different historical roots and different main areas of application. However, they are inter-related and interdependent, and it is not intended that topics would be dealt with in total isolation from each other. Also, while the seven or eight areas, and the contents within each area, are presented in a logical sequence combining, as far as possible, a sensible mathematical order with a developmental one for learners it is envisaged that many content areas listed later in the syllabus would be introduced before or alongside those listed earlier. (For example, geometry appears near the end of the list, but the course committee specifically recommends that introductory geometrical work is started in First Year, allowing plenty of time for the ideas to be developed in a concrete way, and thoroughly understood, before the more abstract elements are introduced.) However, the different order of listing for the Foundation level syllabus does reflect a suggestion that the introduction of some topics (notably formal algebra) might be delayed. Some of these points are taken up in Section 4.
Appropriate pacing of the syllabus content over the three years of the junior cycle is a challenge. Decisions have to be made at class or school level. Some of the factors affecting the decisions are addressed in these Guidelines in Section 4, under the heading of planning and organisation.
3.3 SYLLABUS CONTENT
The contents of the Higher, Ordinary and Foundation level syllabuses are set out in the corresponding sections of the syllabus document. In each case, the content is presented in the two-column format used for the Leaving Certificate syllabuses introduced in the 1990s, with the lefthand column listing the topics and the right-hand column adding notes (for instance, providing illustrative examples, or highlighting specific aspects of the topics which are included or excluded). Further illustration of the depth of treatment of topics is given in Section 5 (in dealing with assessment) and in the proposed sample assessmentmaterials (available separately).
CHANGES IN CONTENT
As indicated in Section 1, the revisions deal only with specific problems in the previous syllabuses, and do not reflect a root-and-branch review of the mathematics education appropriate for students in the junior cycle. The main changes in content, addressing the problems identified in Section 1, are described below. A summary ofall the changes is provided in Appendix 1.
Calculators and calculator-related techniques
As pointed out in the introduction to each syllabus, calculators are assumed to be readily available for appropriate use, both as teaching/learning tools and as computational aids; they will also be allowed in examinations.
The concept of "appropriate" use is crucial here. Calculators are part of the modern world, and students need to be able to use them efficiently where and when required. Equally, students need to retain and develop their feel for number, while the execution of mental calculations, for instance to make estimates, becomes even more important than it was heretofore. Estimation, which was not mentioned in the 1987 syllabus (though it was covered in part by the phrase "the practice of approximating before evaluating"), now appears explicitly and will be tested in examinations.
The importance of the changes in this area is reflected in two developments. First, a set of guidelines on calculators is being produced. It addresses issues such as the purchase of suitable machines as well as the rationale for their use. Secondly, in 1999 the Department of Education and Science commissioned a research project to monitor numeracy-related skills (with and without calculators) over the period of introduction of the revised syllabuses. If basic numeracy and mental arithmetic skills are found to disimprove, remedial action may have to be taken. It is worth noting that research has not so far isolated any consistent association between calculator use in an education system and performance by students from that system in international tests of achievement.
Mathematical tables are not mentioned in the content sections of the syllabus, except for a brief reference indicating that they are assumed to be available, likewise for appropriate use. Teachers and students can still avail of them as learning tools and for reference if they so wish. Tables will continue to be available in examinations, but questions will not specifically require students to use them.
Geometry
The approach to synthetic geometry was one of the major areas which had to be confronted in revising the syllabuses. Evidence from examination scripts suggested that in many cases the presentation in the 1987 syllabus was not being followed in the classroom. In particular, in the Higher level syllabus, the sequence of proofs and intended proof methods were being adapted. Teachers were responding to students' difficulties in coping with the approach that attempted to integrate transformational concepts with those more traditionally associated with synthetic geometry, as described in Section 1.1 of these Guidelines.
For years, and all over the world, there have been difficulties in deciding how indeed, whether to present synthetic geometry and concepts of logical proof to students of junior cycle standing. Their historical importance, and their role as guardians of one of the defining aspects of mathematics as a discipline, have led to a wish to retain them in the Irish mathematics syllabuses; but the demands made on students who have not yet reached the Piagetian stage of formal operations are immense. "Too much, too soon" not only contravenes the principle of learnability (section 2.5), but leads to rote learning and hence failure to attain the objectives which the geometry sections of the syllabuses are meant to address. The constraints of a minor revision precluded the question of "whether" from being asked on this occasion. The question of "how" raises issues to do with the principles of soundness versus learnability. The resulting formulation set out in the syllabus does not claim to be a full description of a geometrical system. Rather, it is intended to provide a defensible teaching sequence that will allow students to learn geometry meaningfully and to come to realise the power of proof. Some of the issues that this raises are discussed in Appendix 2.
The revised version can be summarised as follows.
The approach omits the transformational elements, returning to a more traditional approach based on congruency.
In the interests of consistency and transfer between levels, the underlying ideas are basically the same across all three syllabuses, though naturally they are developed to very different levels in the different syllabuses.
The system has been carefully formulated to display the power of logical argument at a level which hopefully students can follow and appreciate. It is therefore strongly recommended that, in the classroom, material is introduced in the sequence in which it is listed in the syllabus document. For theHigher level syllabus, the concepts of logicalargument and rigorous proof are particularlyimportant. Thus, in examinations, attempted proofsthat presuppose "later" material in order to establish"earlier" results will be penalised. Moreover, proofsusing transformations will not be accepted.
To shorten the Higher level syllabus, only some of the theorems have been designated as ones for which students may be asked to supply proofs in the examinations. The other theorems should still beproved as part of the learning process;students should be able to follow the logical development, and see models of far more proofs than they are expected to reproduce at speed under examination conditions. The required saving of time is expected to occur because students do not have to put in the extra effort needed to develop fluency in writing out particular proofs.
Students taking the Ordinary and (a fortiori)Foundation level syllabuses are not required to prove theorems, but in accordance with the level-specific aims (Section 2.4) should experience the logical reasoning involved in ways in which they can understand it. The general thrust of the synthetic geometry section of the syllabuses for these students is not changed from the 1987 versions.
It may be noted that the formulation of the Foundation level syllabus in 1987 emphasised the learning process rather than the product or outcomes. In the current version, the teaching/learning suggestions are presented in theseGuidelines (chiefly in Section 4), not in the syllabus document. It is important to emphasise that the changed formulation in the syllabus is not meant to point to a more formal presentation than previously suggested for Foundation level students.
Section 4.9 of this document contains a variety of suggestions as to how the teaching of synthetic geometry to junior cycle students might be addressed.
Transformation geometry still figures in the syllabuses, but is treated separately from the formal development of synthetic geometry. The approach is intended to be intuitive, helping students to develop their visual and spatial ability. There are opportunities here to build on the work on symmetry in the primary curriculum and to develop aesthetic appreciation of mathematical patterns.
Other changes to the Higher level syllabus
Logarithms are removed. Their practical role as aids to calculation is outdated; the theory of logarithms is sufficiently abstract to belong more comfortably to the senior cycle.
Many topics are "pruned" in order to shorten the syllabus.
Other changes to the Ordinary level syllabus
The more conceptually difficult areas of algebra and coordinate geometry are simplified.
A number of other topics are "pruned".
Other changes to the Foundation level syllabus
There is less emphasis on fractions but rather more on decimals. (The change was introduced partly because of the availability of calculators though, increasingly, calculators have buttons and routines which allow fractions to be handled in a comparatively easy way.)
The coverage of statistics and data handling is increased. These topics can easily be related to students' everyday lives, and so can help students to recognise the relevance of mathematics. They lend themselves also to active learning methods (such as those presented in Section 4) and the use of spatial as well as computational abilities. Altogether, therefore, the topics provide great scope for enhancing students' enjoyment and appreciation of mathematics. They also give opportunities for developing suitably concrete approaches to some of the more abstract material, notably algebra and functions (see Section 4.8).
The algebra section is slightly expanded. The formal algebraic content of the 1987 syllabus was so slight that students may not have had scope to develop their understanding; alternatively, teachers may have chosen to omit the topic. The rationale for the present adjustment might be described as "use it or lose it". The hope is that the students will be able to use it, and that suitably addressed it can help them in making some small steps towards the more abstract mathematics which they may need to encounter later in the course of their education.
Overall, therefore, it is hoped that the balance between the syllabuses is improved. In particular, the Ordinary level syllabus may be better positioned between a more accessible Higher level and a slightly expanded Foundation level.
CHANGES IN EMPHASIS
The brief for revision of the syllabuses, as described in Section 1.2, precluded a root-and-branch reconsideration of their style and content. However, it did allow for some changes in emphasis: or rather, in certain cases, for some of the intended emphases to be made more explicit and more clearly related to rationale, content, assessment, and via the Guidelines methodology. The changes in, or clarification of, emphasis refer in particular to the following areas.
Understanding
General objectives B and C of the syllabus refer respectively to instrumental understanding (knowing "what" to do or "how" to do it, and hence being able to execute procedures) and relational understanding (knowing "why", understanding the concepts of mathematics and the way in which they connect with each other to form so-called "conceptual structures"). When people talk of teaching mathematics for or learning it with understanding, they usually mean relational understanding. The language used in the Irish syllabuses to categorise understanding is that of Skemp; the objectives could equally well have been formulated in terms of "procedures" and "concepts".
Research points to the importance of both kinds of understanding, together with knowledge of facts (general objective A), as components of mathematical proficiency, with relational understanding being crucial for retaining and applying knowledge. The Third International Mathematics and Science Study, TIMSS, indicated that Irish teachers regard knowledge of facts and procedures as particularly important unusually so in international terms; but it would appear that less heed is paid to conceptual/relational understanding. This is therefore given special emphasis in the revised syllabuses. Such understanding can be fostered by active learning, as described and illustrated in Section 4. Ways in which relational understanding can be assessed are considered in Section 5.
Communication
General objective H of the syllabus indicates that students should be able to communicate mathematics, both verbally and in written form, by describing and explaining the mathematical procedures they undertake and by explaining their findings and justifying their conclusions. This highlights the importance of students expressing mathematics in their own words. It is one way of promoting understanding; it may also help students to take ownership of the findings they defend, and so to be more interested in their mathematics and more motivated to learn.
The importance of discussion as a tool for ongoing assessment of students' understanding is highlighted in Section 5.2. In the context of examinations, the ability to show different stages in a procedure, explain results, give reasons for conclusions, and so forth, can be tested; some examples are given in Section 5.6.
Appreciation and enjoyment
General objective I of the syllabus refers to appreciating mathematics. As pointed out earlier, appreciation may develop for a number of reasons, from being able to do the work successfully to responding to the abstract beauty of the subject. It is more likely to develop, however, when the mathematics lessons themselves are pleasant occasions.
In drawing up the revised syllabus and preparing the Guidelines, care has been taken to include opportunities for making the teaching and learning of mathematics more enjoyable. Enjoyment is good in its own right; also, it can develop students' motivation and hence enhance learning. For many students in the junior cycle, enjoyment (as well as understanding) can be promoted by the active learning referred to above and by placing the work in appropriate meaningful contexts. Section 4 contains many examples of enjoyable classroom activities which promote both learning and appreciation of mathematics. Teachers are likely to have their own battery of such activities which work for them and their classes. It is hoped that these can be shared amongst their colleagues and perhaps submitted for inclusion in the final version of the Guidelines.
Of course, different people enjoy different kinds of mathematical activity. Appreciation and enjoyment do not come solely from "games"; more traditional classrooms also can be lively places in which teachers and students collaborate in the teaching and learning of mathematics and develop their appreciation of the subject. Teachers will choose approaches with whichthey themselves feel comfortable and which meet thelearning needs of the students whom they teach.
The changed or clarified emphasis in the syllabuses will be supported, where possible, by corresponding adjustments to the formulation and marking of Junior Certificate examination questions. While the wording ofquestions may be the same, the expected solutions may bedifferent. Examples are given in Section 5.
3.4 CHANGES IN THE PRIMARY CURRICULUM
The changes in content and emphasis within the revised Junior Certificate mathematics syllabuses are intended, inter alia, to follow on from and build on the changes in the primary curriculum. The forthcoming alterations (scheduled to be introduced in 2002, but perhaps starting earlier in some classrooms, as teachers may anticipate the formal introduction of the changes) will affect the knowledge and attitudes that students bring to their second level education. Second level teachers need to be prepared for this. A summary of the chief alterations is given below; teachers are referred to the revised Primary School Curriculum for further details.
CHANGES IN EMPHASIS
In the revised curriculum, the main changes of emphasis are as follows.
There is more emphasis on
setting the work in real-life contexts
learning through hands-on activities (using concrete materials/manipulatives, and so forth)
understanding (in particular, gaining appropriaterelational understanding as well as instrumentalunderstanding)
appropriate use of mathematical language
recording
problem-solving.
There is less emphasis on
learning routine procedures with no context provided
doing complicated calculations.
CHANGES IN CONTENT
The changes in emphasis are reflected in changes to the content, the main ones being as follows.
New areas include
introduction of the calculator from Fourth Class (augmenting, not replacing, paper-and-pencil techniques)
(hence) extended treatment of estimation;
increased coverage of data handling
introduction of basic probability ("chance").
New terminology includes
the use of the "positive" and "negative" signs for denoting a number (as in +3 [positive three], -6 [negative six] as well as the "addition" and "subtraction" signs for denoting an operation (as in 7 + 3, 24 9)
explicit use of the multiplication sign in formulae (as in 2 ×r , l ×w).
The treatment of subtraction emphasises the "renaming" or "decomposition" method (as opposed to the "equal additions" method the one which uses the terminology "paying back") even more strongly than does the 1971 curriculum. Use of the word "borrowing" is discouraged.
The following topics are among those excluded from the revised curriculum:
unrestricted calculations (thus, division is restricted to at most four-digit numbers being divided by at most two-digit numbers, and for fractions to division of whole numbers by unit fractions)
(Some of these topics were not formally included in the 1971 curriculum, but appeared in textbooks and were taught in many classrooms.)
NOTE
The reductions in content have removed some areas of overlap between the 1971 Primary School Curriculum and the Junior Certificate syllabuses. Some overlap remains, however. This is natural; students entering second level schooling need to revise the concepts and techniques that they have learnt at primary level, and also need to situate these in the context of their work in the junior cycle.
3.5 LINKING CONTENT AREAS WITH AIMS
Finally, in this section, the content of the syllabuses is related to the aims and objectives. In fact most aims and objectives can be addressed in most areas of the syllabuses. However, some topics are more suited to the attainment of certain goals or the development of certain skills than are others. The discussion below highlights some of the main possibilities, and points to the goals that might appropriately be emphasised when various topics are taught and learnt. Phrases italicised are quoted or paraphrased from the aims as set out in the syllabus document. Section 5 of these Guidelines indicates a variety of ways in which achievement of the relevant objectives might be encouraged, tested or demonstrated.
SETS
Sets provide a conceptual foundation for mathematics and a language by means of which mathematical ideas can be discussed. While this is perhaps the main reason for which set theory was introduced into school mathematics, its importance at junior cycle level can be described rather differently.
Set problems, obviously, call for skills of problem-solving; in particular, they provide occasions for logical argument. By using data gathered from the class, they even offer opportunities for simple introduction to mathematical modelling in contexts to which the students can relate.
Moreover, set theory emphasises aspects of mathematics that are not purely computational. Sets are about classification, hence about tidiness and organisation. This can lead toappreciation of mathematics on aesthetic grounds and can help to provide a basis forfurther education in the subject.
An additional point is that this topic is not part of the Primary School Curriculum, and so represents a new start, untainted by previous failure. For some students, therefore, there are particularly important opportunities for personal development.
NUMBER SYSTEMS
While mathematics is not entirely quantitative, numeracy is one of its most important aspects. Students have been building up their concepts of numbers from a very early stage in their lives. However, moving from familiarity with natural numbers (and simple operations on them) to genuine understanding of the various forms in which numbers are presented and of the uses to which they are put in the world is a considerable challenge.
Weakness in this area destroys students' confidence andcompetence by depriving them of theknowledge, skillsand understanding needed for continuing theireducation and for life and work. It therefore handicaps their personal fulfilment and hencepersonaldevelopment.
The aspect of "understanding" is particularly important or, perhaps, has had its importance highlighted with advances in technology.
Students need to become familiar with the intelligent and appropriate use of calculators, while avoiding dependence on the calculators for simple calculations.
Complementing this, they need to develop skills in estimation and approximation, so that numbers can be used meaningfully.
APPLIED ARITHMETIC AND MEASURE
This topic is perhaps one of the easiest to justify in terms of providing mathematics needed for life, workand leisure.
Students are likely to use the skills developed here in "everyday" applications, for example in looking after their personal finances and in structuring the immediate environment in which they will live. For many, therefore, this may be a key section in enabling studentsto develop a positive attitude towards mathematics as avaluable subject of study.
There are many opportunities for problem-solving, hopefully in contexts that the students recognise as relevant.
The availability of calculators may remove some of the drudgery that can be associated with realistic problems, helping the students to focus on the concepts and applications that bring the topics to life.
ALGEBRA
Algebra was developed because it was needed because arguments in natural language were too clumsy or imprecise. It has become one of the most fundamental tools for mathematics.
As with number, therefore, confidence andcompetence are very important. Lack of these underminethepersonal development of the students by depriving them of the knowledge, skills and understanding needed forcontinuing their education and for life and work.
Without skills in algebra, students lack the technical preparation for study of other subjects in school, and in particular their foundation for appropriate studies lateron including further education in mathematics itself.
It is thus particularly important that students develop appropriate understanding of the basics of algebra so that algebraic techniques are carried out meaningfully and not just as an exercise in symbol-pushing.
Especially for weaker students, this can be very challenging because algebra involves abstractions andgeneralisations.
However, these characteristics are among the strengths and beauties of the topic. Appropriately used, algebra can enhance the students' powers of communication, facilitate simplemodelling and problem-solving, and hence illustrate the power of mathematics as a valuablesubject of study.
STATISTICS
One of the ways in which the world is interpreted for us mathematically is by the use of statistics. Their prevalence, in particular on television and in the newspapers, makes them part of the environment in which children grow up, and provides students with opportunities for recognition and enjoyment of mathematics in the worldaround them.
Many of the examples refer to the students' typical areas of interest; examples include sporting averages and trends in purchases of (say) CDs.
Students can provide data for further examples from their own backgrounds and experiences.
Presenting these data graphically can extend students'powers of communication and their ability to shareideas with other people, and may also provide anaesthetic element.
The fact that statistics can help to develop a positiveattitude towards mathematics as an interesting andvaluable subject of study even for weaker students who find it hard to appreciate the more abstract aspects of the subject explains the extra prominence given to aspects of data handling in the Foundation level syllabus, as mentioned earlier. They may be particularly important in promotingconfidence and competence in both numerical and spatial domains.
GEOMETRY
The study of geometry builds on the primary school study of shape and space, and hence relates to mathematics in the world around us. In the junior cycle, different approaches to geometry address different educational goals.
More able students address one of the greatest of mathematical concepts, that of proof, and hopefully come to appreciate theabstractions andgeneralisations involved.
Other students may not consider formal proof, but should be able to draw appropriate conclusions from given geometrical data.
MATHEMATICS
Explaining and defending their findings, in either case, should help students to further their powersof communication.
Tackling "cuts" and other exercises based on the geometrical system presented in the syllabus allows students to develop their problem-solving skills.
Moreover, in studying synthetic geometry, students are encountering one of the great monuments to intellectual endeavour: a very special part of Western culture.
Transformation geometry builds on the study of symmetry at primary level. As the approach to transformation geometry in the revised Junior Certificate syllabus is intuitive, it is included in particular for its aesthetic value.
With the possibility of using transformations in artistic designs, it allows students to encounter creative aspectsof mathematics and to develop or exercise their own creative skills.
It can also develop their spatial ability, hopefully promotingconfidence and competence in this area.
Instances of various types of symmetry in the natural and constructed environment give scope for students' recognition and enjoyment of mathematics in the worldaround them.
Coordinate geometry links geometrical and algebraic ideas. On the one hand, algebraic techniques can be used to establish geometric results; on the other, algebraic entities are given pictorial representations.
Its connections with functions and trigonometry, as well as algebra and geometry, make it a powerful tool for the integration of mathematics into a unified structure.
It illustrates the power of mathematics, and so helps to establish it with students as a valuable subject of study.
It provides an important foundation for appropriatestudies later on.
The graphical aspect can add a visually aestheticdimension to algebra.
TRIGONOMETRY
Trigonometry is a subject that has its roots in antiquity but is still of great practical use to-day. While its basic concepts are abstract, they can be addressed through practical activities.
Situations to which it can be applied for example, house construction, navigation, and various ball games include many that are relevant to the students' life, work and leisure.
It can therefore promote the students' recognition andenjoyment of mathematics in the world around them.
With the availability of calculators, students may more easily develop competence and confidence through their work in this area.
FUNCTIONS AND GRAPHS
The concept of a function is crucial in mathematics, and students need a good grasp of it in order to prepare a firm foundation for appropriate studies lateron and in particular, a basis for further education inmathematics itself.
The representation of functions by graphs adds a pictorial element that students may find aesthetic as well as enhancing their understanding and their abilityto handle generalisations.
This topic pulls together much of the groundwork done elsewhere, using the tools introduced and skills developed in earlier sections and providing opportunities forproblem-solving and simple modelling.
For Foundation students alone, simple work on the set-theoretic treatment of relations has been retained. In contexts that can be addressed by those whose numerical skills are poor, it provides exercises in simple logical thinking.
NOTE
The foregoing argument presents just one vision of the rationale for including the various topics in the syllabus and for the ways in which the aims of the mathematics syllabus can be achieved. All teachers will have their own ideas about what can inspire and inform different topic areas. Their own personal visions of mathematics, and their particular areas of interest and expertise, may lead them to implement the aims very differently from the way that is suggested here. Visions can profitably be debated at teachers' meetings, with new insights being given and received as a result.
The tentative answers given here with regard to whycertain topics are included in the syllabus are, of course, offered to teachers rather than junior cycle students. In some cases, students also may find the arguments relevant. In other cases, however, the formulation is too abstract or the benefit too distant to be of interest. This, naturally, can cause problems. Clearly it would not be appropriate to reduce the syllabus to material that has immediately obvious applications in the students' everyday lives. This would leave them unprepared for further study, and would deprive them of sharing parts of our culture; in any case, not all students are motivated by supposedly everyday topics.
Teachers are therefore faced with a challenging task in helping students find interest and meaning in all parts of the work. Many suggestions with proven track records in Irish schools are offered in Section 4. As indicated earlier, it is hoped that teachers will offer more ideas for an updated version of the Guidelines. |
A Level Maths Core 2 Collins Student Support Materials for Edexcel AS Maths Core 2 covers all the content and skills your students will need for their Core 2 examination, including: * Algebra and functions * Coordinate geometry in the (x, y) plane * Sequences and series * Trigonometry * Exponentials and logarithms * Differentiation * Integration * EXAM PRACTICE * Answers |
Pre-Algebra
"Glencoe Pre-Algebra" is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique ...Show synopsis"Glencoe Pre-Algebra" is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique language of mathematics, so that they are prepared for every type of problem-solving and assessment situation |
Computer algebra system
A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.
The expressions typically include polynomials in multiple variables; standard functions of expressions (sin, exponential, etc.); arbitrary functions of expressions; integrals, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. (This is a recursive definition.)
The symbolic manipulations supported typically include
simplification
substitution of symbolic or numeric values for expressions
change of form of expressions: expanding products and powers, rewriting as partial fractions, etc.
Many also include a high level programming language, allowing users to implement their own algorithms.
The study of algorithms useful for computer algebra systems is known as computer algebra.
The run-time of numerical programs implemented in computer algebra systems is normally longer than that of equivalent programs implemented in systems such as MATLAB, GNU Octave, or directly in C, since they are programmed for full symbolic generality and thus cannot use machine numerical operations directly.
History
Computer algebra systems began to appear in the early 1970s, and evolved out of research into artificial intelligence, though the fields are now regarded as largely separate. The first popular systems were Reduce, Derive, and Macsyma which are still commercially available; a copyleft version of Macsyma called Maxima is actively being maintained. The current market leaders are Maple and Mathematica; both are commonly used by research mathematicians, scientists, and engineers. MuPAD is a commercial system, also available in a free version with slightly restricted user interface for non-commercial research and educational use. Some computer algebra systems focus on a specific area of application; these are typically developed in academia and |
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Class1
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Class January 9, 2008 To Do for Friday, January 11, 2008 1. Read Sections 5.6, Integration by Parts 2. Complete this handout to turn in. You may do this handout using your text or notes. You may discuss general differentiation and integration strategies with others, but you are to do the problems on your own without looking at anyone else's work. This handout represents some of the topics and techniques 174, Winter 2000 1. Answer the following questions: a) b) c) d) e)Initial: _What are the primary advantages of double heterostructures (DH) for lasers? Why is the absorption region of a typical Si photodiode much thicker than that of a typical
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Calculus II Class 2, January 11, 2008 Topic:Section 5.5: Substitution and Section 5.6: Integration by Parts Important Concepts: Symmetry in integrals. Importance of the "constant of integration". The rule for integration by parts. Which function
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To introduce some methods for approximating a function by polynomials, splines and
rational functions.
To introduce the concept of best approximation.
To introduce Gaussian quadrature rules.
Intended
Learning Outcomes:
On successful completion of this module students will:
Have acquired active knowledge and understanding of basic approximation theory
(including the formulation of some best approximation problems) and of an advanced
technique in numerical integration.
Be able to approximate a function by a polynomial, piecewise polynomial or Padé
approximant.
Pre-requisites:
153, 157, 211 (ex-UMIST),
MT1202 Sequences and Series (ex-VUM)
This course is NOT available to students who
have taken MT2181 in previous years.
Dependent Courses:
Course Description:
The first half of the course is concerned with approximation
theory. The aim is to approximate a complicated function by a much simpler function (such
as a polynomial), which is easier to evaluate, differentiate and integrate. In numerical
integration, the course builds on the ideas introduced in Module 157. A family of
integration rules, known as Gaussian quadrature rules, is introduced. |
A Reference Guide to Vector Algebra is an introductory textbook on vectors that is written for high school AP and first year college students in math, physics and engineering. Included with the... More...
A Reference Guide to Vector Algebra is an introductory textbook on vectors that is written for high school AP and first year college students in math, physics and engineering. Included with the book on CD-ROM are The Vector Algebra Tools; a suite of interactive software tools. Specifically, The Vector Algebra Tools are a comprehensive set of vector algebra calculators that enable the algebraic determination and manipulation of vector quantities in a Cartesian coordinate system.
Therefore, whether you are involved in the classroom or independent study study assignments, preparing for a test, or solving a specific problem, A Reference Guide to Vector Algebra will help you acquire the necessary problem solving skills, strategies, and confidence to solve a wide variety of vector algebra problems.
Science Tap, Inc. is an educational software company that develops multimedia applications in the physical sciences for high-school and college students.
For more information please visit are web site |
Welcome to Algebra 2! There are no secrets to success. It is the result of preparation, hard work and learning from failure -Colin Powell- Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures So enter the doors of --- and lets explore the Wide World of Algebra 2. No need to guess; know your current grade or what you have missed by logging on to PINNACLE. Need your graduation status then VIRTUAL COUNSELOR is your information gateway. Need the latest news from your school? Why not log on to Miramar High and be informed. |
Mathematical Applications in Agriculture the specialized math skills you need to be successful in today's farming industry with MATHEMATICAL APPLICATIONS IN AGRICULTURE, 2nd Edition. Invaluable in any area of agriculture-from livestock and dairy production to horticulture and agronomy--this easy to follow book gives you steps by step instructions on how to address problems in the field using math and logic skills. Clearly written and thoughtfully organized, the stand-alone chapters on mathematics involved in crop production, livestock production, and financial management allow you... MORE to focus on those topics specific to your area while useful graphics, case studies, examples, and sample problems to help you hone your critical thinking skills and master the concepts. Invaluable in any area of agriculture or as a hands-on learning tool in introductory math courses, the 2nd Edition of MATHEMATICAL APPLICATIONS IN AGRICULTURE demonstrates industry-specific methods for solving real-world problems using applied math and logic skills students already have. |
**The statistics class invites you to take a quick on-line survey about your music listening habits. Thank you for participating!
Statistics 2011-12 Instructor: Eric Rhomberg
How can we understand and communicate about numbers and data that describe real, practical situations?
How can we best prepare ourselves in terms of mathematics to be successful in our post-high school lives?
How can we increase our comfort and confidence level in mathematics?
These will be the essential questions that guide our work throughout the year. This course is for students who want an alternative to pre-calculus and calculus – students who would benefit more from solidifying basic skills, preparing to be "college ready" in terms of computational skills, and developing their practical math fluency and confidence.
In this course, we will:
Use Statisitics as a playing field for developing our overall math fluency.
Review math skills and concepts from basic computations through algebra and geometry.
Play math games, solve puzzles, engage in problem solving challenges and "number talks" in order to develop our "math minds" and our fluency with numbers.
Support your peers. Do your part to create a safe and effective collaborative environment.
Do 20 minutes of focused math homework each night. If you get stuck on a problem, always write it out as far as you can take it (even if that just means writing out the initial problem). Bring your homework to class everyday, and be ready to go over it and ask questions.
Complete projects by the due date. Communicate with the instructor in advance to negotiate extensions.
Try to have fun with all of this!
The flow of the class:
Our class periods will vary depending on what we are working on. We will often start class with a "Warm-up" problem or challenge to activate our math thinking.
We will then divide the period up between two or three of the following:
Direct skill instruction to the whole class.
Individualized work and practice, with instructor(s) doing 1-on-1 coaching.
Games and Puzzles that liven things up and develop our math fluency.
Group problems that emphasize collaboration.
"Number Talks" in which we really break down and articulate how we think about and solve math problems. |
logarithms, fractions, fraction/decimal conversions, degrees to decimal conversion, reciprocals, factorials, grads, hyperbolics, polar/rectangular conversions. Fraction feature allows operations with fractions and mixed numbers. Two-variable statistics allow you to enter, delete, insert, and edit individual statistical data elements. Calculator runs on solar and battery power.
TI-34 MultiView is ideal for middle school math, pre-Algebra, Algebra I and II, trigonometry, general science, geometry, and biology. MultiView display shows fractions as they are written on paper. View multiple calculations on a four-line display and easily scroll through entries. Enter multiple calculations to compare results and explore patterns on the same screen. Simplify and convert fractions to decimals and back again. Integer division key expresses results as quotient and remainders. Toggle Key lets you quickly view fractions, decimals and terms including Pi in alternate forms. Functions include previous entry, power, roots, reciprocals, variable statistics and seven memories. Scientific calculator also features user-friendly menus, automatic shutoff, hard plastic color-coded keys, nonskid rubber feet, impact-resistant cover with a quick-reference card, and dual power with solar and battery operation.
Handheld graphing calculator lets you visualize in dynamic graphing with the high-resolution full-color, backlit display. Ideal for Algebra, Trigonometry, Geometry, Statistics, Business/Finance, Biology, Physics, Chemistry, and Engineering. Touchpad navigation works more like a laptop computer. You can even transfer class assignments from this handheld to your PC or Mac computer. Calculator also lets you color-code equations, objects, points and lines and make faster, stronger connections between equations, graphs and geometric representations on screen. 3D Graphing lets you graph and rotate manually and automatically. The thin, lightweight TI-Nspire CX runs on the included rechargeable battery and includes a wall adapter, software, a USB Unit-to-unit cable, and USB Unit-to-Computer cable. |
Lecture 1: Simple Equations
Embed
Lecture Details :
Introduction to basic algebraic equations of the form Ax=B
Course Description :
This is the original Algebra course on the Khan Academy and is where Sal continues to add videos that are not done for some other organization. It starts from very basic algebra and works its way through algebra II. |
Even though contemporary biology and mathematics are inextricably linked, high school biology and mathematics courses have traditionally been taught in isolation. But this is beginning to change. This volume presents papers related to the integration of biology and mathematics in high school classes.
The first part of the book provides the rationale for integrating mathematics and biology in high school courses as well as opportunities for doing so. The second part explores the development and integration of curricular materials and includes responses from teachers.
Papers in the third part of the book explore the interconnections between biology and mathematics in light of new technologies in biology. The last paper in the book discusses what works and what doesn't and presents positive responses from students to the integration of mathematics and biology in their classes.
Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with the Association for Computer Machinery (ACM).
Readership
High school teachers, education specialists, graduate students, and research mathematicians interested in mathematics and biology education. |
Beginning Algebra With Applications
9780618803590
ISBN:
0618803599
Pub Date: 2007 Publisher: Houghton Mifflin
Summary: Intended for developmental math courses in beginning algebra imm...ediate feedback, reinforcing the concept, identifying problem areas, and, overall, promoting student success."New!" "Interactive A Concepts of Geometry section has been added to Chapter 1."New!" Coverage of operations on fractions has been changed in Section 1.3 so that multiplication and division of rational numbers are presented first, followed by addition and subtraction"New!" A Complex Numbers section has been added to Chapter 11, "Quadratic Equations.""New Media!" Two key components have been added to the technology package: HM Testing (powered by Diploma) and, as part of the Eduspace course management tool, HM Assess, an online diagnostic assessmentEl Monte, CAShipping:StandardComments: 0618803599 MULTIPLE COPIES AVAILABLE. New book may have school stamps but never issued. 100% gua... [more] 0618803599 MULTIPLE COPIES AVAILABLE. New book may have school stamps but never issued. 100% guaranteed fast shipping! ! |
Description
Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.
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Illustrating the Fundamental Theorem
of Calculus Using Spreadsheets
Daniel Lotesto and Susan Thurmond
Introduction
The Fundamental Theorem of Calculus, which states
,
is one of the most difficult concepts for a student to grasp. Students do not always see the relationship between the area bounded by y = f(x), the x-axis, x=a, and x=b, and the difference between the antiderivative values F(b) and F(a). This spreadsheet activity illustrates the theorem by using Euler's Method to approximate the value for F(b) - F(a) and Riemann Sums to approximate the definite integral. The graphs offer a visual representation of the theorem.
Preliminary Discussion of Euler's Method
Given a function F(x), and its derivative = F'(x) = f(x) , we can approximate function values for F assuming that we know initial values (x0, y0) where y0 = F(x0). Choosing a value for x, we can generate values of x and y using the following recursive equations:
A particularly useful function to use for the derivative is F'(x) = f(x) = since, in most courses, F(x) = ln x has not yet been defined as
.
Therefore, not only are students unburdened by preconceptions of the graph of y = F(x) while building the spreadsheet, they actually construct the graph of y = ln x in the process.
Activity
Working in pairs or small groups, have the students prepare spreadsheets to generate y values using Euler's Method and to find the areas of the corresponding rectangles using the left endpoint rule. The amount of instructor input will depend upon the ability level of the class. Sample spreadsheets using Microsoft Excel are shown in this article as well as a brief description of how to generate them. They are offered as a guide. The instructor is encouraged to work with her/his students to create their own version.
Spreadsheet #1 uses Euler's Method to find the approximate y values for y = F(x). The initial values for x0 and y0 are entered in cells [A12] and [D12], respectively. The value for x is entered in [C5]. The following formulas are used to complete the spreadsheet:
[A13] =A12+$C$5, fill down
[B13] =1/A12, fill down
[C13] =B13*$C$5, fill down
[D13] =D12+C13, fill down
[E12] =D12 - $D$12, fill down
Spreadsheet #2 is a continuation of spreadsheet #1. It uses the left endpoint rule to compute areas of rectangles. The formulas are as follows:
[F7] =A12
[F8] =F7+$C$5, fill down
[G7] =1/F7, fill down
[H7] =G7*$C$5, fill down
[I7] =H7
[I8] =I7+H8, fill down
[K6] =E12, fill down
SPREADSHEET #1
A
B
C
D
E
1
Constructing the Graph of F(x) Using Euler's Method
2
3
Given: f(x) = slope function (the derivative of some function F(x))
4
Given: f(x) = 1/x; F(a)=F(1)=0
5
Given x =
0.1
6
7
8
F(x)
9
x
f(prev x)
f(prev x)* x
y = prev y +
F(x)-F(a)
10
f(prev x)* x
11
12
1
0
0
13
1.1
1
0.1
0.1
0.1
14
1.2
0.90909091
0.090909091
0.190909091
0.19090909
15
1.3
0.83333333
0.083333333
0.274242424
0.27424242
16
1.4
0.76923077
0.076923077
0.351165501
0.3511655
17
1.5
0.71428571
0.071428571
0.422594073
0.42259407
18
1.6
0.66666667
0.066666667
0.489260739
0.48926074
19
1.7
0.625
0.0625
0.551760739
0.55176074
20
1.8
0.58823529
0.058823529
0.610584269
0.61058427
21
1.9
0.55555556
0.055555556
0.666139824
0.66613982
22
2
0.52631579
0.052631579
0.718771403
0.7187714
23
2.1
0.5
0.05
0.768771403
0.7687714
24
2.2
0.47619048
0.047619048
0.816390451
0.81639045
25
2.3
0.45454545
0.045454545
0.861844996
0.861845
26
2.4
0.43478261
0.043478261
0.905323257
0.90532326
27
2.5
0.41666667
0.041666667
0.946989924
0.94698992
28
2.6
0.4
0.04
0.986989924
0.98698992
29
2.7
0.38461538
0.038461538
1.025451462
1.02545146
30
2.8
0.37037037
0.037037037
1.062488499
1.0624885
31
2.9
0.35714286
0.035714286
1.098202785
1.09820278
32
3
0.34482759
0.034482759
1.132685544
1.13268554
Figure 2
SPREADSHEET #2
F
G
H
I
J
K
Approximating Area Bounded by f(x), the x-axis, x = a and x = b
Using Riemann Sums and the Left-Endpoint Rule
Left endpoint
Height (H)
A=H* x
Areas
F(x)-F(a)
0
1
1
0.1
0.1
0.1
1.1
0.90909091
0.09090909
0.19090909
0.19090909
1.2
0.83333333
0.08333333
0.27424242
0.27424242
1.3
0.76923077
0.07692308
0.3511655
0.3511655
1.4
0.71428571
0.07142857
0.42259407
0.42259407
1.5
0.66666667
0.06666667
0.48926074
0.48926074
1.6
0.625
0.0625
0.55176074
0.55176074
1.7
0.58823529
0.05882353
0.61058427
0.61058427
1.8
0.55555556
0.05555556
0.66613982
0.66613982
1.9
0.52631579
0.05263158
0.7187714
0.7187714
2
0.5
0.05
0.7687714
0.7687714
2.1
0.47619048
0.04761905
0.81639045
0.81639045
2.2
0.45454545
0.04545455
0.861845
0.861845
2.3
0.43478261
0.04347826
0.90532326
0.90532326
2.4
0.41666667
0.04166667
0.94698992
0.94698992
2.5
0.4
0.04
0.98698992
0.98698992
2.6
0.38461538
0.03846154
1.02545146
1.02545146
2.7
0.37037037
0.03703704
1.0624885
1.0624885
2.8
0.35714286
0.03571429
1.09820278
1.09820278
2.9
0.34482759
0.03448276
1.13268554
1.13268554
Figure 3
Sample Questions to Explore:
1. How do the values in the columns labeled SArea and Approx F(x) - F(a) compare? What does this suggest to you?
2. Let a = 1.6. Without changing your spreadsheet, what is the sum of the areas from a to xn = b = 2.7? [Hint: the left-endpoint of the last interval is 2.6, not 2.7]. What is the value of F(b) - F(a)? Show on your graphs what these two numbers represent.
3. Change Dx to 0.05. [C5]. Does SAreas = Approx F(x) - F(a)?
4. Change Dx to 0.01. [C5]. Result?
5. Change Dx back to 0.1. Change the interval from [1,3] to [0.5, 2.5] by making the following changes: x0 = 0.5 [A12] ; y0 = -.6931 [D12]. These two values must be given. How do the values in the last two columns compare? Write out what these two values represent. [Note: a rectangle lying under the x-axis indicates a "negative area."]
6. Extend your spreadsheet to include 40 intervals so that the domain of x is [0.5, 4.5]. What is the equation for y = F(x)? Using your calculator, test your "guess".
7. Explore the function f(x) = -cos x from x0 = a = 0.5 to xn = b = 2.5. Change row 4 to read f(x) = -cos x ; F(a) = F(0.5) = -.4794. Change the following cells: [C5] to 0.1; [A12] to 0.5; [D12] to -.4794; [B13] to = -cos(A12), fill down; and [G7] to = -cos(F7), fill down. What is the approximate value of F(2.5) - F(0.5)? What is the value of Areas? Look at your graphs. Are they what you expect? Is the graph of F(x) the graph of the antiderivative of f(x) = -cosx?
Note to the instructor: Below is an analytical verification of what the comparison of the two spreadsheets illustrates. [Kevin Bartkovich, NCSSM]
f(x) is the derivative of some function F(x). The y values refer to points on the graph of the curve that approximates F(x) using Euler's Method. The point (x0,y0) is the initial point. |
College Algebra-enhanced Edition - 6th edition
Summary: Accessible to students and flexible for Accessible to students and flexible for instructors, COLLEGE ALGEBRA, SIXTH EDITION, uses the dynamic link between concepts and applications to bring mathematics to life. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, work independently, and obtain greater mathematical fluency. The text also includes technology features to accommodate courses that allow the option of using graphi...show moreng calculators. Additional program components that support student success include Eduspace tutorial practice, online homework, SMARTHINKING Live Online Tutoring, and Instructional DVDs. The authors' proven Aufmann Interactive Method allows students to try a skill as it is presented in example form. This interaction between the examples and Try Exercises serves as a checkpoint to students as they read the textbook, do their homework, or study a section. In the Sixth Edition, Review Notes are featured more prominently throughout the text to help students recognize the key prerequisite skills needed to understand new |
College Algebra - 2nd edition
ISBN13:978-0201347111 ISBN10: 0201347113 This edition has also been released as: ISBN13: 978-0201383980 ISBN10: 0201383985
Summary: The Dugopolski Precalculus series for 1999 is technology optional. With this approach, teachers will be able to w...show moreho will study additional mathematics, this text will provide the skills, understanding and insights necessary for success in future courses. For those students who will not pursue further mathematics, the extensive emphasis on applications and modeling will demonstrate the usefulness and applicability of mathematics in today's world. Additionally, the focus on problem solving that is a hallmark of this text provides numerous opportunities for students to reason and think their way through problem situations. The mathematics presented here is interesting, useful, and worth studying. One of the author's principal goals in writing this text was to get students to feel the same way. New! Linking Concepts This new feature is located at the end of nearly every exercise set. It is a multipart exercise or exploration that can be used for individual or group work. The idea of this feature is to use a concept from the current section along with concepts from previous sections, and ask questions that help students see the links among various concepts. Some parts of these questions are open-ended, and require somewhat more thought than standard exercises. Answers to this feature are given only in the Instructor's Solutions Manual.
New! Applications Hundreds of new exercises have been added to the exercise sets, most based on and involving applications of real-world situations. The emphasis of the new exercises is on understanding concepts and relationships.
New! Exercise Sets The exercise sets have been examined carefully to ensure that the exercises range from easy to challenging, and are arranged in order of increasing difficulty. Many new exercises require a graphing calculator.
New! Regression Problems Many new regression problems have been included in the text, so that students can start with real data, and use a calculator to obtain mathematical models of real problem situations.
New! Graphing Calculator Exercises Optional exercises that require a graphing calculator are now located in more natural positions in the exercises rather than at the end of the exercise sets as in first edition. The exercises are optional and are marked with a graphing calculator icon.
New! Graphing Calculator Discussions Optional graphing calculator discussions have been integrated into the text, and are set off with graphing calculator icons so that they can be easily skipped by those not using this technology.
New! Web Site A new Web site has been established that is designed to increase student success in the course by offering section-by-section tutorial help, enhancement of text group projects, downloadable TI programs and author tips. An icon alerts students to when this site would be useful. The site will also be helpful to instructors by providing useful resources for teaching a precalculus course.
Chapter Opener Each chapter begins with a Chapter Opener that discusses a real-world situation in which the mathematics of the chapter is used. Examples and exercises that relate back to the opener are included within the chapter.
Index of Applications The many applications contained within the text are listed in an Index of Applications that immediately follows the Table of Contents. The applications are page referenced and grouped by subject matter.
For Thought Each exercise set begins with a set of true or false questions that review the basic concepts in that section, help check student understanding before beginning the exercises, and offer opportunities for writing and/or discussion.
Highlights This end-of-chapter feature presents an overview of each section of the chapter and is a useful summary of the basic information that students should have mastered in that chapter.
Chapter Review Exercises These exercises are designed to review the chapter, without reference to the individual sections, and prepare students for the Chapter Test.
Chapter Test The problems in the Chapter Test are designed to help students measure their readiness for a classroom test, and instructors may use them as a model for their own end of chapter tests.
Tying It All Together This is a review of selected concepts from the present and prior chapters, and requires students to integrate multiple concepts and skills.
Content Changes
Revised Chapter P This chapter contains prerequisite material on real numbers, rules of exponents, factoring, and simplifying expressions. Basic linear, quadratic, and absolute value equations and inequalities are covered in Chapter 1. Some sections from both of these chapters may be omitted depending on the preparation level of the students.
New! Revised Chapter 3 Quadratic type equations, equations with rational exponents or radicals, and more complicated absolute value equations now occur in Section 3.5, following the theory of polynomial equations, Section 3.4. Because some of these equations are polynomial equations, they will be better understood after the theory of polynomial equations has been studied.
New! Parametric Equations A new section on parametric equations has been added to Chapter 7.
New! Vector Dot Products Material on vector dot products has been added to the coverage of vectors in Chapter 7GreenEarthBooks Portland, OR
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Description as mathematics, while others are more suitable for students majoring in mathematics. This book concentrates on clear explanations with plenty of examples. Most of the problems have fully worked answers.
Topics included are:
basic algebra
set theory
functions
introductory calculus
multivariate calculus
financial mathematics
linear algebra.
Table of contents
Author biography
Penelope de Boer (BSc Maths, Canterbury) is a trained teacher who has taught at St Bede's College in Christchurch. After time out to raise four children she returned to teach at Otago Polytechnic. She now teaches first-year courses in quantitative studies and economics at Victoria University where she is the 100-level Programme Director in the School of Economics and Finance. She has previously published Maths Made Easy (Pearson Education, 2002), a book for those who have never enjoyed maths, nor been good at it, but recognise that they need to understand basic mathematics in order to progress in their careers.
Mohammed Khaled (BA Dhaka, MSc (Econ) Islam, MA (Econ) Essex, PhD Br Col.) has been teaching mathematics to students at the Faculty of Commerce and Administration at Victoria University since 1998, but he has been teaching economics here since 1985. He also taught economics as a visiting teacher at the Universities of British Columbia, Hawaii and Sydney. His teaching philosophy is to back up applications with a theoretical understanding of why things work in the simplest possible way. Since he also teaches microeconomics and econometrics, the mathematical linkages to these and other subjects taken by commerce students are considered carefully in his teaching of mathematics. |
An Introduction to Manifolds (Universitext)
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
Beginning C# is a book that offers a lot of guidance, in a format that readers find intuitive to follow. The exercise-based format of the Wrox Beginning series has a strong following by all levels of ...
An introduction to basic ideas in differential topology, based on the many years of teaching experience of both authors. Among the topics covered are smooth manifolds and maps, the structure of the ...
Applied Behavior Analysis (ABA) is based on the premise that behavior can be influenced by changes in environment and by the reinforcing consequences of that behavior. This introductory guide to ABA ...
Although Scripture cannot be reduced to a mere piece of human writing, there is much to gain by paying careful attention to the historical, literary, and theological dimensions of the biblical text. ... |
"…to
be possessed of a vigorous mind is not enough; the prime requisite
is rightly to apply it. The greatest minds, as they are capable
of the highest excellences, are open likewise to the greatest aberrations;
and those who travel very slowly may yet make far greater progress,
provided they keep always to the straight road, than those who,
while they run, forsake it."
Rene Descartes
(1596-1650) Discourse
on the Method of Rightly Conducting the Reason, and Seeking Truth
in the Sciences
General
Requirements for Licensure in Mathematics:
Completion
of the College requirements for graduation including the core curriculum
requirements.
Completion
of the requirements for a mathematics major.
Candidates for mathematics licensure are required to complete
Mathematics 210: Mathematical Modeling mathematics
licensure must submit to the Department of Education a copy of their
major project completed for Mathematics 210: Mathematical
Modeling.
Standards for Mathematics Teachers
The
following standards are mandated by the North Carolina Department of
Public Instruction and are imbedded in the specialty area
coursework.
Teacher
candidates possess the mathematical knowledge needed to enable students to
understand numbers, ways of representing numbers, and relationships among
numbers and number systems and to enable students to understand meanings
of operations and how they relate to one another.Candidates enable students to
develop computational fluency and to make reasonable estimates.At the middle and secondary grade
levels, teacher candidates need the mathematical knowledge to enable
students to transfer their understanding of numbers and number operations
to symbolic expressions involving variables.
Understand
and apply the mathematics of natural, integer, rational, real, and
complex number systems.
Understand
and apply the mathematics of algebraic structures (e.g. groups, rings
and fields) and rules for operations on expressions, equations,
inequalities, vectors and matrices.
Demonstrate
skill in using algebra to model real-world
applications.
Standard 2:Spatial sense, measurement and
geometry
Teacher
candidates possess the mathematical knowledge needed to enable students to
analyze the characteristics and properties of 2- and 3-dimensional
geometric shapes; to develop mathematical arguments about geometric
relationships; to understand units, processes of measure, and measurable
attributes of objects; and to apply appropriate techniques, tools, and
formulas to determine measurements.They enable students to develop the visualization, spatial
reasoning, and geometric modeling to solve problems.Teacher candidates particularly at
middle and secondary grade levels need the mathematical knowledge to
enable students to use coordinate geometry in solving problems, to
understand concepts of symmetry, and to apply
transformations.
Understand
core concepts and principles of Euclidean geometry in the plane and
space.
Use
axiomatic reasoning and demonstrate facility with
proof.
Understand
and apply the use of coordinates in 2- and 3-dimensional geometry,
vectors and transformations, including matrix representations of
transformations.
Understand
and apply trigonometry from a geometric perspective and demonstrate
skill in using trigonometry to solve problems.
Standard 3:Patterns, relationships, and
functions
Teacher
candidates possess the mathematical knowledge needed to enable students to
understand patterns, relations, and functions.This includes the use of algebraic
symbols to represent and analyze mathematical situations, the use of
mathematical models to represent and understand quantitative
relationships, and the analysis of "change" in various contexts.
Understand
and recognize patterns in data that are modeled by important classes of
functions.
Understand
and perform transformations of functions by arithmetically combining,
composing, and inverting.
Demonstrate
and apply knowledge of important classes of functions (e.g., polynomial,
exponential and logarithmic, rational, and periodic), including the
effect of changing parameters within these classes of
functions.
Teacher
candidates possess the mathematical knowledge needed to enable students to
formulate questions that can be addressed with data, along with the
necessary skills to collect, organize, and display relevant data to answer
those questions.They enable
students to select and use appropriate statistical methods to analyze
data, to understand and apply basic concepts of probability, and to
develop and evaluate inferences and predictions that are based on
data.
Engage
in data investigations, including formulating questions and collecting
data to answer questions.
Understand
and use standard techniques for organizing, displaying and analyzing
univariate data, with the ability to detect patterns and departures from
patterns.
Teacher
candidates possess the mathematical knowledge needed to enable students to
develop skills in problem solving, making connections between various
branches of mathematics, reasoning and proof, and communication and
representation of mathematical ideas.
Use
algebraic reasoning effectively for problem solving and proof in number
theory, geometry, discrete mathematics, and
statistics.
Judge
the reasonableness of numerical computations and their results.
Judge
the meaning, utility, and reasonableness of the results of symbolic
manipulations, including those carried out by
technology.
Standard 6:Mathematical tools
Teacher
candidates must be versed in the appropriate use of mathematical tools and
manipulatives. |
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space andMultivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involveIn Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally This notation is well-suited to describing systems of differential equations, and taking derivatives of matrix-valued functions with respect to matrix variables. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of theIn Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change This notation is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data.Engineering is the Discipline and Profession of applying technical and scientific Knowledge andIn Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notationalPhysics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.
Notice
This article uses another definition for vector and matrix calculus than the form often encountered within the field of estimation theory and pattern recognition. Estimation theory is a branch of Statistics and Signal processing that deals with estimating the values of parameters based on measured/empirical dataPattern recognition is a sub-topic of Machine learning. It is "the act of taking in raw data and taking an action based on the category of the data" The resulting equations will therefore appear to be transposed when compared to the equations used in textbooks within these fields.
Notation
Let M(n,m) denote the space of realn×m matrices with n rows and m columns, whose elements will be denoted F, X, Y, etc. In Mathematics, the real numbers may be described informally in several different ways An element of M(n,1), that is, a column vector, is denoted with a boldface lowercase letter x, while xT denotes its transpose row vector. In Linear algebra, a column vector or column matrix is an m × 1 matrix, iThis article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a An element of M(1,1) is a scalar, and denoted a, b, c, f, t etc. All functions are assumed to be of differentiability classC1 unless otherwise noted. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability
Vector calculus
Because the space M(n,1) is identified with the Euclidean spaceRn and M(1,1) is identified with R, the notations developed here can accommodate the usual operations of vector calculus. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner
Matrix calculus
For the purposes of defining derivatives of simple functions, not much changes with matrix spaces; the space of n×m matrices is after all isomorphic as a vector space to Rnm. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalarIn Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents theSuppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in someIn Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The three derivatives familiar from vector calculus have close analogues here, though beware the complications that arise in the identities below.
The tangent vector of a curve F : R → M(n,m)
The gradient of a scalar function f : M(n,m) → R
Notice that the indexing of the gradient with respect to X is transposed as compared with the indexing of X. The directional derivative of f in the direction of matrix Y is given by
where tr denotes the trace. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal
The differential or the matrix derivative of a function F : M(n,m) → M(p,q) is an element of M(p,q) ⊗M(m,n), a fourth ranktensor (the reversal of m and n here indicates the dual space of M(n,m)). History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventuallyHistory The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventuallyIn Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In short it is an m×n matrix each of whose entries is a p×q matrix.
and note that each ∂F/∂Xi,j is a p×q matrix defined as above. Note also that this matrix has its indexing transposed; m rows and n columns. The pushforward along F of an n×m matrix Y in M(n,m) is then
Note that this definition encompasses all of the preceding definitions as special cases.
Identities
Note that matrix multiplication is not commutative, so in these identities, the order must not be changed. In Mathematics, commutativity is the ability to change the order of something without changing the end result
Examples
Derivative of linear functions
This section lists some commonly used vector derivative formulas for linear equations evaluating to a vector. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions.In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable
Derivative of quadratic functions
This section lists some commonly used vector derivative formulas for quadratic matrix equations evaluating to a scalar.
Derivative of matrix traces
This section shows examples of matrix differentiation of common trace equations. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive lengthIn Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal
Relation to other derivatives
There are other commonly used definitions for derivatives in multivariable spaces. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change For topological vector spaces, the most familiar is the Fréchet derivative, which makes use of a norm. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis.In Mathematics, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalizeIn Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In the case of matrix spaces, there are several matrix norms available, all of which are equivalent since the space is finite-dimensional. In Mathematics, a matrix norm is a natural extension of the notion of a Vector norm to matrices. However the matrix derivative defined in this article makes no use of any topology on M(n,m). Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. It is defined solely in terms of partial derivatives, which are sensitive only to variations in a single dimension at a time, and thus are not bound by the full differentiable structure of the space. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constantIn Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential For example, it is possible for a map to have all partial derivatives exist at a point, and yet not be continuous in the topology of the space. See for example Hartogs' theorem. NB that the terminology is inconsistent and Hartogs' theorem may also mean Hartogs' lemma on removable singularities or the result on Hartogs number In Mathematics The matrix derivative is not a special case of the Fréchet derivative for matrix spaces, but rather a convenient notation for keeping track of many partial derivatives for doing calculations, though in the case that a function is Fréchet differentiable, the two derivatives will agree.
Usages
Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. In mathematical optimization problems the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of This includes the derivation of:
Alternatives
The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. The Kalman filter is an efficient Recursive filter that estimates the state of a Dynamic system from a series of noisy measurementsIn Signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notationalIn Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. Note that a matrix can be considered simply a tensor of rank two.
See also
External links
Matrix calculus appendix from Introduction to Finite Element Methods book on University of Colorado at Boulder. Derivative is a fundamental construction of Differential calculus and admits many possible generalizations within the fields of Mathematical analysis, Combinatorics Uses the Hessian (transpose to Jacobian) definition of vector and matrix derivatives. |
Math and Science Skills Center
The Math & Science Skills Center is a unique resource which provides assistance in all levels of Rhodes State Mathematics and Science Courses to our students at NO COST. The Skills Center, staffed by a Coordinator, Professional and Peer Tutors who provide the basis for the tutoring services. Each time students come to the Skills Center for assistance, they must complete the following steps to receive tutoring services:
Step One:Log in their name using student ID card on any of the 2 computers next to the entrance.
Step Two:If the student signs in as SELF STUDY then he/she does not need our help but wants to do independent work. Else the next available peer tutor or I will welcome him/her and assist with whatever he/she needs help with.
Step Three: Students must log out on the Skills Center computer when leaving. This helps the MSSC monitor and improve its services. We look forward to your visits.
Hours of Operation and Location:
Hours: Monday to Thursday 8am to 8pm and Friday 8am to 5pm. We are closed on weekends. Location: 240 Science Building
No appointments are necessary - students are encouraged to drop-in anytime.
The Mathematical and Science Quotes
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." - John Louis von Neumann.
"Mathematics is well and good but nature keeps dragging us around by the nose." -Albert Einstein. |
Mathematics for Engineering: An
ASL Qualification for the Advanced Diploma in
Engineering
Introduction
Mathematics is an integral
part of the study of engineering regardless of which
branch of engineering is chosen. Many in the
engineering community believe that additional
mathematics material should be available for those
students studying the Advanced Diploma in
Engineering to prepare them for progression onto
engineering degree courses at university. They also
appreciate that teachers in schools and colleges
need more real engineering examples to underpin the
essential mathematics and also to excite interest in
engineering. In response to these challenges,
members of the engineering and maths communities
joined together in May 2007 to form a Maths Task
Group.
Maths Task Group
The group contains
representatives from well established organisation
like
Observers from the
Qualifications and Curriculum Development Agency and
a number of awarding organisations have also
contributed to the work of the Task Group.
The group has developed an
Additional and Specialist Learning (ASL) Mathematics
qualification that is available for any Level 3
learner wishing to develop their mathematical
skills and knowledge in a real life context,
especially in engineering. A consensus is emerging
that students thinking of studying engineering at
university should take this qualification as an ASL
qualification along with the Advanced Diploma in
Engineering at Level 3. For more information please
download the flyer below:
This ASL qualification is
based on a foundation year course taught at
Loughborough University over many years. This course
is designed for students without A level Mathematics
who wish to go on to study engineering to degree
level. It has been designed to contain all the
necessary topics from A level Mathematics to
facilitate such study. Very good results in their
subsequent engineering studies have been achieved by
students at Loughborough who followed this course.
This ASL qualification
provides an appropriately rigorous maths programme
within the Engineering Diploma, tailored to the
needs of engineering students. To provide powerful
motivation to students, teachers/lecturers will be
expected to highlight practical engineering
applications of the mathematics in the course. We
are confident, based on the Loughborough experience,
that success in this demanding programme can prepare
students on the Engineering Diploma for subsequent
university studies in all branches of Engineering,
and possibly other science or technology subjects as
well.
To support the teachers of
this qualification, a number of Engineering
Mathematics Exemplars have been developed during the
last two years and they are available to download
from this webpage; please see below.
The exemplars are intended to:
motivate mathematics
teaching and learning
provide support for
teachers teaching contextualised mathematics for
the first time
help students gain fluency
in the use of mathematics for practical problem
solving
illustrate the
applicability of the mathematics in the ASL unit
exemplify valuable
activities undertaken by engineers
Proposed Distribution of 50 exemplars over different engineering streams
Newton's laws of motion; Hooke's law of
elasticity; Solving simultaneous equations using
the matrix method followed by the substitution
method that involves first and second
derivatives; Expanding a determinant and finding
inverse of a matrix
Newton's Laws of motion and Hooke's Law of
stiffness, Principle of Conservation of
Momentum, Product rule of differentiation,
Finding maximum values using first derivatives,
Plotting graphs using Excel or similar software
We are committed to produce as many exemplars as possible
in any of the above Engineering Stream. Further support in exemplar development will
be highly appreciated. If you have any further question, wish to add your
suggestions, need more information or want to support this development with an
engineering case study, please contact:
Dr Sapna Somani
Project Officer (Engineering) - National HE STEM
Programme
The Royal Academy of Engineering
3 Carlton House Terrace
London
SW1Y 5DG
The Royal Academy of
Engineering, in collaboration with the Institute of
Mathematics and its Applications (IMA), has produced
some exciting career videos filming young and
diverse engineers working in industries, showing how
mathematics is a vital part of engineering in their
day to day life. |
Appropriate for upper level undergraduate and graduate courses in Mathematical Modeling offered in math, engineering departments, and applied math departments. Prerequisite is some exposure to differential equations and to matrices. This accessible and practical text is designed to nurture a "m... |
Mathematical Communication
Balancing conceptual with formal
Students of mathematics often have trouble appropriately deciding when to build conceptual understanding and when to present ideas formally; additionally, students often don���t realize that conceptual explanations must be as carefully constructed as formal presentations. This webpage provides sample recitations, writing exercises, proofs, and presentations that can be used to help students understand how to effectively build conceptual understanding, how to present mathematics formally and when to do each. |
only text that takes basic math concepts and applies them specifically to HVAC! This unique text covers the entire range of mathematical problems and subjects encountered by HVAC technicians in real-world situations. With practice problems, a review unit, and three review tests, students can easily assimilate the material presented and visualize its use in the field. A glossary defines terms specific to HVAC, while conversion charts present critical field information at a glance. This book works well as a math text all by itself, or in conjunction with a general math text. An instructor's guide includes two achievement review tests with answer keys, as well as answer keys to problems in the book.
ALSO AVAILABLE
INSTRUCTOR SUPPLEMENTS CALL CUSTOMER SUPPORT TO ORDER
Instructor's Manual, ISBN: 0-8273-7949-8 |
Mathematics
All students will be recommended for the math course most appropriate to their skill level and needs.
2.
Math students who fail the first semester may be moved to a more appropriate math course for their level or work habits.
3.
Math students who fail the second semester of a course may not move on to a more advanced course until they repeat and pass the course. No partial credit is given for having passed only one semester of a year course.
4.
Any student who wishes to retake a course for a higher grade may do so, but must take the course in the next consecutive year. Both grades will remain on the transcript, but credit will be granted only once.
5.
Any student who has earned an Algebra 1 credit may not enroll in Algebra 1A or Algebra 1B.
6.
Any exception to the above will require administrative approval.
Graphing Calculators
The Sugarcreek Local School district is committed to expanding, integrating, and utilizing technology within the total school environment. Graphing calculators represent one example of such technology. In addition to other higher level mathematical applications, these calculators allow students to easily graph, interpret, and compare even the most difficult linear, polar, parametric, and statistical functions.
As the number of various makes and models of graphing calculators has grown, it has become increasingly difficult for the teachers of mathematics to remain proficient with the functionality of all makes and models available. Therefore, it has become necessary for Sugarcreek Local Schools to standardize on one particular calculator. For the following reasons, the Texas Instruments TI-83/84 family of calculators was selected. This includes any of the following: TI-84+ Silver Edition, TI-84+, TI-84, TI-83+ Silver Edition, TI-83+, TI-83, and the new TI-Nspire.
The TI-83/TI-84 calculators provide functionality consistant with the areas covered by the Sugarcreek Local Schools curriculum for mathematics.
The TI-83/TI-84 calculators provide functionality allowing for use beyond high school in most non-specialized college/university level math classes.
The TI-83/TI-84 calculators were found to be easier to use and more cost effective than other graphing calculators having comparable functionality.
The TI-Nspire is the newest in the family of TI calculators. While it offers numerous new features and expanded functionality, it also comes with a switcable face plate that allows it to provide the same functionality as the TI-83/84 calculators. So that our students can benefit from the added functionality that the Nspire has to offer, the Mathematics Department will begin to transition to the TI-Nspire during the 08-09 school year. Students purchasing a new calculator, specifically those taking advanced classes, are encouraged to purchase a TI-Nspire. However, students already owning a TI-83/84 calculator and/or those who don't wish to purchase a TI-Nspire may continue to use the TI-83/84 for most required courses.
Each student has the option to purchase and use a calculator other than those listed above for homework and day-to-day activity. However, it must be recognized that the student's teacher may not be able to answer questions regarding the use or functionality of calculators other than those listed. |
Mathematics
The Mathematics department at St John Fisher aspires to the highest standards of excellence in teaching and learning for all our students.
We will be a source for the promotion of problem solving, analytical thinking and utilizing technology. We will produce high quality mathematical thinking students that are well prepared to enter the work force or aspire to higher education.
We live in a mathematical world where if we decide on a purchase, choose an insurance or health plan, or use a spreadsheet we rely on mathematical understanding. The World Wide Web, CD-Roms and other media disseminate vast quantities of quantitative information. The level of mathematical thinking and problem solving needed in the workplace has increased dramatically over the years.
In such a world those who understand and can do mathematics will have opportunities that others do not. Mathematical competence opens doors to productive futures. A lack of mathematical competence closes those doors.
Students have different abilities, needs and interests yet everyone needs to use mathematics in his or her personal life, workplace and in further study. At St John Fisher we believe that all students deserve an opportunity to understand the power and beauty of mathematics and embrace it into their lives. |
MATLAB & Simulink Student Version 2012a
Description
Get the essential tools for your courses in engineering, math, and science.
MATLAB® is a high-level language and interactive environment that lets you focus on your course work and applications, rather than on programming details. It enables you to solve many numerical problems in a fraction of the time it takes to write a program in a lower-level language such as Java™, C, C++, or Fortran. You can also use MATLAB to analyze and visualize data using automation capabilities, thereby avoiding the manual repetition common with other products.
The MATLAB in Student Version provides all the features and capabilities of the professional version of MATLAB software, with no limitations. There are a few small differences between the Student Version interface and the professional version of MATLAB:
The MATLAB prompt in Student Version is EDU>>
Printouts contain this footer: Student Version of MATLAB
For more information on this product please visit the MathWorks website:
IMPORTANT NOTE:Proof of student status is required for activation of license
Features
Contains R2012a versions of:
MATLAB
Simulink
Symbolic Math Toolbox
Control System Toolbox
Signal Processing Toolbox
Signal Processing Blockset
Statistics Toolbox
Optimization Toolbox
Image Processing Toolbox
Student Version also comes with complete user documentation on the CD Rom.
IMPORTANT NOTE:Proof of student status is required for activation of license
New to this Edition
The 2012a release includes 2 important new features:
Target Hardware support directly from Simulink: Of special interest to educators is the addition of Simulink Control Design and Simulink features to enable project-based learning. Student Version now includes built-in Windows support to run Simulink models on low-cost target hardware, including LEGO MINDSTORMS NXT and BeagleBoard. With a click your model moves from simulation onto hardware, further increasing your return on Model-Based Design with Simulink |
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 |
MAT 125: Calculus A
About this course: The goal of this course is to develop your understanding of the concepts of
Calculus and your ability to apply them to problems both within and outside of Mathematics. Functions
are presented and analyzed as tables, graphs, and formulas. You need to continue to develop your
proficiency at manipulating formulas and equations, which are the language of science. Fluency in this
language is essential for success in science or engineering.
Text:Single Variable Calculus (Stony Brook Edition), by James Stewart. This is the same book as
Stewart's Concepts and Contexts, 4th edition, but with a different cover and a lower price. This same
book is used by MAT 125, MAT 126, MAT127, MAT131 and MAT132; Suffolk Community College also
uses this book, but with the other cover.
Calculators: You may find using a graphing calculator (TI 83, TI84 are the best choices) helpful. Some
of the HW problems will require a calculator for their solution but no quiz or test questions will require
the calculator. Also, in this course, no calculators will be allowed on exams.
Homework: You cannot learn calculus without working problems. Expect to spend at least 8 hours a
week solving problems; do all of the assigned problems, as well as additional ones to study. If you do
not understand how to do something, get help from your TA, your lecturer, your classmates, or in the
Math Learning Center (in the basement of the Mathematics Tower). Almost every lecture will include
homework. We will be using WebAssign, a web-based system in which you see the problems, submit
your answers and/or solutions and get immediate feedback on your work. You will be graded on how
many questions you get correct and how many tries it takes you to get the correct answer. You will
receive more information concerning its use in separate documents. These will be posted in Bb as they
become available. Keep an eye on your email for announcements.
Recitations: These are required sessions with your TA in which homework will be discussed and
questions will be answered. Tests and quizzes will be returned during recitations.
Quizzes: The grades you receive by submitting your HW through WebAssign will be half of the
recitation grade. Quizzes given in class will be the other half.
Reading: The textbook is intended to be read. Read the assigned sections corresponding to the
assignments.This will greatly increase your comprehension, and enable you to ask intelligent questions
in class. Furthermore, the lectures will not always be able to cover all of the material for which you will
be responsible.
Examinations and grading: There will be two evening exams, and the ever-popular final exam. The
dates and times are listed below; the locations will be announced in lecture. Success on the exams will
require correct and efficient solutions to the more difficult of the homework problems.
Math Learning Center: The Math Learning Center, in Math S-240A, is there for you to get help with
Calculus. It is staffed most days and some evenings -- your lecturer or TA may hold some of his or her
office hours there. A schedule should be posted outside the room and at the Math Undergraduate
Office.
Cell Phones: If you have a cell phone with you, please have it turned off or set to vibrate. Allowing your
phone go off in class is inconsiderate. If you need to talk to someone on the phone please leave the
room and move far enough away so that the class is not disturbed. Playing video games on the phone
and text messaging are equally inappropriate.
DSS advisory: If you have a physical, psychological, medical, or learning disability that may affect
your course work, please contact Disability Support Services (DSS) office: ECC (Educational
Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with
you what accommodations are necessary and appropriate. Arrangements should be made early in the
semester (before the first exam) so that your needs can be accommodated. All information and
documentation of disability is confidential. Students requiring emergency evacuation are encouraged to
discuss their needs with their professors and DSS. For procedures and information, go to the following
web site and search Fire safety and Evacuation and Disabilities.
Conduct: The following statement is University policy:
"Stony Brook University expects students to maintain standards of personal integrity that are in harmony with the
educational goals of the institution; to observe national, state, and local laws and University regulations; and to
respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial
Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning
environment, and/or inhibits students' ability to learn."
Email: Please be sure to use Standard English in writing emails to me, your lecturer and TA, including
correct spelling and punctuation. Also use complete sentences. Start the subject line with "MAT 125"
then your subject.
Blackboard: You are required to use this application throughout the course to access assignments and
other material, to view grades, to contact me and to check for announcements. (See the box below.) It
is also a good way to contact other students in class to complain about the professor. It would best if
you check the site and your email daily since you are responsible for any updates or other material that
are posted.
You can access class information on-line at: If you have used Stony
Brook's Blackboard system previously, your login information (Username and Password) has not
changed. If you have never used Stony Brook's Blackboard system , your initial password is your
SOLAR ID# and your username is the same as your Campus Net ID , which is generally your first
initial and the first 7 letters of your last name (if you have a sparky account, it's your username). |
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Math Workbook for ISEE, SSAT, and HSPT Prep Middle and High School Entrance Exams
This second edition of the Math Workbook for ISEE, SSAT, & HSPT Prep: Middle School and High School Entrance Exams has been overhauled from the first edition to reflect the most up-to-date knowledge of the private school admissions exams, as well as to incorporate new insights gleaned by our experts as they used the first edition to prepare students for these exams.
Here are some new features you will find in the second edition:* A more logical progression of concepts and exercises* Over 60 new practice sets covering basic arithmetic, algebra, geometry, and advanced topics.* Expanded sections specific to the ISEE and HSPT* An assignment planner to help students track their practice sets and measure scores* A formula sheet containing the most vital math rules and information* A thorough explanation of the major differences between the ISEE, SSAT, and HSPT* Updated answer key with easier navigationThe philosophy of this math workbook remains the same as in our first edition; rigor and drill. Because these are the first tests that actively try to trick students at every turn, those who sit for these exams need reflexive familiarity with mathematical computation , problem types, and strategy. The entrance exams are the first standardized tests for which budgeting time is a significant issue. Students need to spend the majority of time on analysis, rather than computation, to avoid getting tricked. By building skills, speed, and confidence, we hope to eliminate anxiety and give students a solid foundation on which to build excellent scores. This book is intended as a supplement for our highly trained staff, so it does not include strategies. However, motivated students can use it successfully with occasional help from a teacher or parent. Each chapter is comprised of units, with each unit comprised of problem sets with difficulty increasing in a logically progressive manner. Students should do as many of the problem sets for each unit as it takes to achieve a 90% accuracy rate. As a general rule, students taking lower level exams should complete chapters 1-8, and stick to "basic" questions in chapters 9-16. Students preparing for high school entrance exams should go through the entire book.While private school entrance exam preparation is the primary purpose of this book, we recognize that it may serve other purposes as well. This book would be useful for anyone looking for a workbook that encompasses all fundamental math concepts up through an 8th grade math program.For further information about the book and our test prep offerings, check out our website at Education (c) 2012
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List price:
$25.00
Edition:
N/A
Publisher:
CreateSpace Independent Publishing Platform
Binding:
Trade Paper
Pages:
538
Size:
8.25" wide x 10.75" long x 1.25" tall Math Workbook for ISEE, SSAT, and HSPT Prep Middle and High School Entrance Exams - 9781475143225 at TextbooksRus.com. |
"But why do I need math?" Now, using this series, give students a clear, definitive, and logical answer. Math is necessary to get the job done in most technical fields, including auto mechanics, electricity/electronics, and the building trades. Each video shows real-life problem situations solved by using practical math and actual computations on the screen. Use Introduction to Math in Technology as an overview and then progress to specific topics. At last...a program to help your students succeed in the world of technical math.Reading a Ruler: English and Metric Measurements In the first lesson, the different forms of English measurement are discussed and displayed as they would appear on a ruler. The viewer also learns how to understand fractions when measuring and how to find exact measurements using a ruler. The secon...(more details)
DVD
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3-Year Streaming
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Area and Volume This video describes how to calculate the area of rectangles and other shapes-both geometrical and irregular-and how to determine the volume of a rectangular solid. Dramatized segments and computer animations focus on calculating lawn dimensions at a...(more details)
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$69.95
DVD + 3-Year Streaming
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3-Year Streaming
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Surface Area and Volume Whether wallpapering a footlocker or filling a cylinder with corncobs, a knowledge of three-dimensional shapes is essential. This program demystifies the subjects of surface area and volume by sharing solid information backed up by the surface area f...(more details)
DVD
$99.95
DVD + 3-Year Streaming
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3-Year Streaming
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Measurement This video describes how to estimate costs of products and services, determine the circumference of an object and its effect on motion, and calculate area and volume. Dramatized segments and computer animations illustrate ways to use measurements tak...(more details)
DVD
$69.95
DVD + 3-Year Streaming
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3-Year Streaming
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Units, Perimeter, Circumference, and Area When it comes to measuring flat shapes, geometry generously provides a formula for every occasion. This program begins with an overview of how to convert English and metric units of measurement. Next, finding the perimeter of polygons is illustrated,...(more details) |
Search Course Communities:
Course Communities
Lesson 9: Gaussian Reduction
Course Topic(s):
Developmental Math | Systems of Equations
An introduction to solving a 3x3 linear system of equations. Back substitution and triangular form are first introduced, and then a general procedure for solving systems is presented. Application and non application problems. |
Mathematics
Campbell High School provides a curriculum that aims to promote a positive attitude towards mathematics and its relevance to students both personally and in their community. The program gives students the opportunity to develop a sound understanding of mathematics and its processes.
Years 7 and 8
A common course of study is provided to students in years 7 and 8, which is structured to accommodate the range of students in each cohort. Provision is also made for students in years 7 and 8 who need additional assistance with key elements of the course. An advanced class exists in year 7 to extend the gifted students. The year 7 and 8 courses cover the basic skills of mathematics and enable students to be numerate in our society as well as preparing them for further mathematical enquiry.
Years 9 and 10
In years 9 and 10, all students study a common core program. This program provides students with opportunity to study mathematics in different levels of depth. Students can follow a level 1, level 2 or level 3 program of study. All courses are divided into semester units and cover core topics, option topics and assessment.
Level 1 and 2 coursescover topics which prepare students for tertiary mathematics at college.
The level 1 program leads to a further study of mathematics as needed in heavily mathematics based college courses such as Physics, Chemistry and Mathematical Methods. High achieving level 1 students have an opportunity to select Extension Mathematics as an elective in years 9 and 10. This course provides a head start for students likely to attempt the top mathematical stream at college.
The level 2course allows students the opportunity to deepen their understanding of fundamental mathematical skills as well as providing an opportunity to develop their mathematical literacy.
The level 3course takes on a more thematic approach and teaches students how to apply mathematical knowledge.
Mathematics Tutorials
A formalised mathematics tutorial session is available to all students through Enrichment. It is designed to give those students who require extra help in mathematics the opportunity to work through problems with a teacher in a smaller group.
Mathematics Competitions
Students at Campbell High School have the opportunity to compete in a number of Mathematics competitions including the Australian Informatics Competition, the Australian Mathematics Competition and the Australian Mathematics Challenge. Students who have an interest in the creative understanding of mathematics can also join the mathematics and engineering section of the Tournament of Minds. |
These videos are intended as educational materials for my VVC students. You MAY download the files for your PERSONAL educational use only. You may NOT change, edit or distribute these video files in any way. You may NOT upload these files to any web location without the express written permission of Stephen Toner.
You will need to have Real Player installed to view some of the video lectures. After following the link, click on the words "Get RealPlayer Free" in the Basic Player column.
Summary of all of the methods of factoring in Introductory Algebra. Common mistakes that should be avoided. This is a rather long video. You might want to right-click and choose "save as" to store this to your hard drive.
How to solve quadratic equations by factoring. Word problems involving quadratic equations. This is a rather long video. You might want to right-click and choose "save as" to store this to your hard drive. |
Sixth Form
We offer a range of courses in Mathematics at sixth form level. Below is some brief information about our courses. To find out more please use the sixth form section of the sidebar to the right of the page.
Further Mathematics AS / A2
This can be taken in addition to
the single Mathematics A-level. The AS-level is particularly suitable for
students intending to study Mathematics, Engineering, Computer Studies or
Physics as it contains an introduction to more advanced pure maths topics, more
discrete maths and mechanics. It is no more difficult than the single maths AS.
The A2 is suitable for the better mathematicians as it extends and develops the
topics covered in the single A-level. Good mathematicians will really enjoy it
(the teaching staff certainly do). Students will find taking two maths A-levels
less demanding and so it is very suitable as a fourth or even fifth A-level.
Some Further Maths modules can be used for single maths to improve overall
grades.
Use of Mathematics AS
The pilot GCE Use of Mathematics
is distinct from GCE Mathematics and is designed to encourage more students to
continue their mathematics studies post-16. Students now have the opportunity to pursue
practical and relevant mathematics courses to the same level as traditional GCE
Mathematics.
Use
of Mathematics, and the constituent Free-Standing Mathematics Qualifications
(FSMQ) courses of which it is composed, were developed to enable the study of
mathematical topics in practical, real-life contexts. As Professor Adrian Smith
stated in his 2004 report into Mathematics 14-19, students involved in FSMQ
courses recognise the relevance of the mathematics as they model the real world
and develop skills which are readily transferable to either the real world or
to their other studies.
Mathematics AS / A2
The aim of this syllabus has been
to produce a course which, while challenging, is accessible and enjoyable to
all students. The course develops ability and confidence in mathematics and its
applications, together with an appreciation of how mathematical ideas help in
an understanding of the world and the society in which we live. It also extends
the G.C.S.E. teaching and assessment methods into the sixth form. |
Pre-Algebra Essentials For Dummies
Paperback
Click on the Google Preview image above to read some pages of this book!
Many students worry about
starting algebra. Pre-Algebra Essentials For Dummies provides
an overview of critical pre-algebra concepts to help new algebra
students (and their parents) take the next step without fear. Free of
ramp-up material, Pre-Algebra Essentials For Dummies contains
content focused on key topics only. It provides discrete explanations
of critical concepts taught in a typical pre-algebra course, from
fractions, decimals, and percents to scientific notation and simple
variable equations. This guide is also a perfect reference for parents
who need to review critical pre-algebra concepts as they help students
with homework assignments, as well as for adult learners headed back
into the classroom who just need to a refresher of the core concepts.
The Essentials For Dummies Series
Dummies is proud to present our new series, The Essentials For
Dummies. Now students who are prepping for exams, preparing to
study new material, or who just need a refresher can have a concise,
easy-to-understand review guide that covers an entire course by
concentrating solely on the most important concepts. From algebra and
chemistry to grammar and Spanish, our expert authors focus on the
skills students most need to succeed in a subject.
About The Author
Mark Zegarelli is a math tutor and author of several books, including Basic Math & Pre-Algebra For Dummies.
Reviewed By Toni Whitmont, Booktopia Buzz Editor
To read more reviews by Toni Whitmont, click here to visit the Booktopia Newsletter Archive.
We all know and love the Dummies series. Now students who are prepping for exams, preparing to study new material, or who just need a refresher can have a concise, easy-to-understand review guide that covers and entire course by concentrating solely on the most important concepts. From alegbra and chemistry to grammar and Spanish, these expert authors focus on the skills students most need to succeed in a subject.
This Essentials series is perfect for final year school students, people doing bridging courses for university and mature age students needing a refresher. |
cheap scientific calculator that does matrix operations
cheap scientific calculator that does matrix operations
I do not have a calculator that does matrix stuff that would be great for my classes this semester. I don't want to spend $100+ on a graphing calculator since I use matlab at home strictly now. Im looking to solve simultaneous equations, matrix operations like det,eigen,mult,...etc
There is an excellent software calculator called Mathwizard that does Matrices,algebra,calculus,scientific calculator and plot graph.check it out. you can also find mobile applications that does matrices,algebra,calculus,differential equations , and plot graphs |
Skill Development Notes
This site area literally sketches what
may done with set skills and concepts. Email the site author for further
help.
Modern pure mathematics from counting to calculus may
be formed and expressed in terms of sets and set theory assumptions and
practices. The modern mathematics course designs of the 1950's
emphasized a introductory form of set theory and notation. Most
elements of mathematics were presented in terms of sets. Those course
designs lingers today in mathematics education as is or diluted. The
dilution was evident when I taught senior high school courses whose
content required set concepts to students whose earlier education did
not cover sets.
The question of when to introduce set skills and
concepts further depends on school systems. Just before employed may be
a good time.
Set concepts [Venn
Diagrams and subset builder notation] may be useful in illustrating and
developing logic mastery for the development of mathematics and to aid
precision in reading and writing outside of mathematics. In counting for
its own sake or the calculation of probabilities, sets and functions may
be employed to track and count items, ways or possibilities. And
probability itself may be precisely expressed and calculated with the aid
of set concepts.
The set development of functions and relations is covered in full and
clarified in the site more algebra area on function. Course
designers need to decide how to much cover in senior high school
mathematics and calculus, and how. Show the coverage be all at once. Or,
should the coverage be woven as needed into the study of further
functions and computation rule required by college programs. In
preparation of students for college pograms in technical fields, fields
that employ calculus, a lean coverage of the set and graphic
representation of function may be sufficient with this representation
being one of many until or if the study of pure mathematics begins.
The verbal description of of generalized commutative and associative
laws for arithmetic may be easily described using sets. The expansion
of product
of sums in terms of sums of products of their terms may also be
described with sets The
exercises here check them while numerically hinting at
equivalent computation rules. |
Specific course objectives
The course is aimed at providing the basic knowledge of calculus (differential,
integral) for real and vector-valued functions of one or several real variables, together with the fundamentals of the theory of series and with an introduction
to ordinary differential equations. Lectures will be mainly focused on the
comprehension of notions (definitions, results), although some proofs will still be detailed. Examples and exercises will be presented.
By the end of the course the Students are expected to be able to handle
correctly and without hesitation limits, derivatives, function graphs,
integrals (also multiple, line and surface integrals), series, linear differential equations, and the corresponding theoretical facts.
Differential Equations.
Introduction to ordinary differential equations. The Cauchy problem.
Linear ordinary differential equation of the first order. Linear ordinary differential equation of the second order with constant coefficients.
Line Integrals and Surface Integrals.
Curves: tangent vectors; rectifiable curves and arc length. Surfaces: tangent planes; surface area. Line integrals with respect to the arc length. Line integrals of vector fields, and applications in Physics. Gradient fields, potentials, and path independence. The operators curl and div. Surface integrals, and applications in Physics. Green's theorem and divergence theorem in two variables. Stokes' theorem and divergence theorem in three variables.
Course entry requirements
Mathematics: the required prerequisites for enrollment into the Engineering
Faculty.
Testing and exams
Finals consist in a written (exercises) and an oral test (theory plus possibly
exercises). Both written and oral tests have to be passed within the same final
session. In order to be admitted to the oral test, a specific minimum of points has to be obtained in the written test. |
MAT
315
- Abstract Algebra
Algebra is concerned with sets of objects and operations on these sets. This course will take students beyond the real number and polynomials to groups and other algebraic structures. In a modern, or abstract algebra course, one assumes a small number of basic properties as axioms and then proves many other properties from the axioms. This will assist the student in becoming more proficient at proof-writing. |
Create interactive Maple applications that provide a simple point-and-click interface to the full power of Maple, from solving an integral or visualizing a function to computing the dissolved oxygen content in a river or calculating the necessary thickness of foil for a solar panel. These interactive math apps can be used to solve problems, visualize solutions, explore concepts, and assess understanding.
Using Maple's Smart Document interface, you can create Möbius Apps quickly and easily using the Exploration Assistant, or take advantage of Maple's flexible development environment by using drag-and-drop interactive components to create more customized applications. Möbius Apps can contain interactive elements such as sliders, buttons, and math entry boxes, as well as text, mathematics, plots, and images that explain or enhance the application.
Share it
Share the application with everyone
Share your applications with everyone through The Möbius Project. You, your colleagues, and your students can use Möbius Apps to enhance lectures, enrich assessment, and support independent learning.
To make your App available to everyone, including non-Maple users, simply post it to the Maple Cloud from inside Maple. Möbius Apps can be used from a computer, tablet, or smart phone from within your browser. MapleNet is available 24/7 to provide the mathematics engine behind your Möbius Apps. If you prefer, you can also download the free Maple Player to run Möbius Apps right on your own computer. Of course, if you do have Maple, you can download these Apps from the Maple Cloud straight into your Maple session.
Grade it
Grade the application to assess understanding
Möbius Apps are gradable, so you can use them to measure and deepen your students' understanding. By embedding a Möbius App into a Maple T.A. assignment question, it can be used as a specialized calculator for working out steps of the solution or visualizing the problem, or the Möbius App itself can be the question. To answer the question, students interact with the App directly inside Maple T.A.
Maple T.A. will automatically grade the Möbius App by looking at what the student did with it, such as examining the location of sliders and the contents of math entry boxes. Like other Maple T.A. questions, if there is more than one correct answer, Maple T.A. will analyze the response mathematically to assign an appropriate grade. |
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
DescriptionFeatures & benefits
The Access to Stage 5.1 Maths 9 Teacher'sResource (printout and CD) provides a range of photocopiable worksheets for each chapter: remedial or consolidation worksheets, review worksheets, language worksheets, activities, games, problems and group activities. In addition, there is a teaching program, support for assessment and answers to the worksheets and Homework Book. The Teacher's Resource CD includes a PDF of both the printout and the student coursebook, a link to the companion website and spreadsheet activities.
Recommended CD specifications
PC
Intel® Pentium® III processor or higher
Windows® XP or Vista
Microsoft® Office 2000, XP or 2007
128MB or more of installed RAM (256MB recommended for complex forms or large documents) |
This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhed |
Computer Science Facilities
Pearsons Hall, Room 110
The department of mathematics and computer science runs a lab devoted to math and computing. The lab contains 11 dual-core workstations which run specialized mathematical and computer science software alongside standard publishing and spreadsheet programs. Mathematics software includes Maple and Octave, and the computer science software includes development environments for Java, C++, Scheme, Python, and many other programming languages.
We encourage math and computer science majors to learn LATEX (a program for typesetting mathematics documents), so our lab workstations run WinEdit, a text editor designed to work with LATEX. The lab also houses a Linux server and an eight node Beowulf cluster built as a joint project between students and faculty. The lab's location is Pearsons 110; it is open to all Drury students from 8 a.m. to midnight each weekday.
Next door to the lab is a lounge for mathematics and computer science majors. The lounge contains a dual-core workstation, a whiteboard, a sofa, a refrigerator, a microwave, a coffeemaker, and a television. |
Course Goals and Philosophy
The purpose of this course is to revisit the content
of the elementary mathematics curriculum with the focus on
understanding the underlying concepts and justifying the solutions of
problems dealing with this material. The focus is not on being able to
perform the computations (the how to do it), although that is a
necessity as well, but on demonstrating an ability to explain
why you can solve the problem that way or why the
algorithm works that way. You will need to be able communicate your
explanations
both verbally and in writing with strict attention to the mathematical
accuracy and clarity of your explanation. You will have the chance
to work with mathematical concepts in an active, exploratory manner
as recommended by the National Council of Teachers of Mathematics
(NCTM):
Knowing mathematics means being able to use it in
purposeful ways.
To learn mathematics, students must be engaged in exploring,
conjecturing, and thinking rather than only rote learning of rules and
procedures. Mathematics learning is not a spectator sport. When
students construct knowledge derived from meaningful experiences, they
are much more likely to retain and use what they have learned. This
fact underlies the teacher's new role in providing experiences that
help students make sense of
mathematics, to view and use it as a tool for reasoning and problem
solving.
If you feel a need to review elementary school
mathematics, this is your responsibility. For this purpose, I
recommend reading our textbook and consulting with me outside of
class. For a reference on the content of elementary school
mathematics,
I have provided a summary of the NCTM standards
.
It is also the purpose of this
course to improve your ability to engage in mathematical thinking
and reasoning, to increase your ability to use mathematical knowledge
to solve problems, and to learn mathematics through problem
solving.
The emphasis in this course is on learning numerical mathematical
concepts through solving problems. You will often work with other
students for the following reasons: Group problem solving is
often
broader, more creative, and more insightful than individual
effort.
While working on problems with others, students practice putting their
mathematical ideas and reasoning into words. This ability to
explain
mathematics is clearly essential for future teachers. While
working
in groups, students learn to depend on themselves and each other
(rather
than the instructor) for problem solutions. In groups, students
can
motivate each other to excel and accept more challenging
problems. Motivation to persevere with a difficult problem may be
increased.
Socialization skills are developed and practiced. Students are
exposed to a variety of thinking and problem-solving styles different
from their
own. Interaction with others may stimulate additional insights
and
discoveries. Conceptual understanding is deeper and
longer-lasting when ideas are shared and discussed.
Grading
Your grade in this course will be based upon your
performance on problem sets, weekly questions, reflection logs, three
exams, and the final project. A possible weight assigned to each
is designated on the left in the grade definition scale given on the
right:
20% - Problem
Sets
A 90 - 100
10% - Weekly
Questions
B 80 - 89.99
10% - Each
of two In-Class Exams
C 70 - 79.99
15% - Final
Project
D 60 - 69.99
15% - Comprehensive
Final Exam
E 0 - 59.99
10% - Reflection
Logs
10% - Participation
If you would like your grade to use a different
weighting scale, please inform me by Friday, September 9. In
addition, you must pass several Basic Skills Checks
throughout the
semester or your course grade will be lowered by a half
letter (e.g. from a B to a B-) for each incomplete check. Further
details are available below.
Participation
You are preparing to enter a profession where good
attendance is crucial and expected. It is important that you make
every attempt to attend class, since active involvement is an integral
part
of this course. Since much of the class is experiential, deriving
the
same benefits by merely examining someone's class notes or reading the
textbook would be impossible. Each class period you will be
working on activities with your group. If you are working in your
group you will receive
one participation point that day. If you also participate to the
class as a whole (answer a question, present a solution, ask an
insightful question or offer important relevant commentary) you will
receive two participation points for that day. If you are not
working in your group, you will receive no points for that day.
Working each day and never speaking in class will earn 80%.
Speaking every other day on which there is
an opportunity to speak will earn 95%. Scores between will be
scaled
linearly.
Problem Sets
There will be several problem sets throughout the
semester. These will consist of problems from the text. The goal
of these assignments is to have you practice
solving problems and then being able to write clear, detailed, and
mathematically accurate solutions that explain what you did and why.
Simple numeric answers with some math computations (or work) shown will
not be sufficient. There will be an in-class discussion of the
difference between a "solution" and an "answer". Assignments are due at
the start of class and must be easy to read. Late assignments will not
be accepted.
These questions and papers will be graded on the
following scale
Question
(out of 4)
0 – missing question
1 – question copied
2 – partial question
3 – completed question (with some
solution)
4 – completed question correctly
and well-written
Assignments will be returned on the following class day.
Weekly Questions
On Wednesdays, I will assign a question relating to
the topic for the previous week. They will be due approximately
once a month as indicated on the schedule. The goal of these
assignments is for you to write substantial explanations of the main
concepts presented in class. They will eventually be incorporated
into your final project. Before the final project, they will be
collected for completeness and marked with suggestions.
Assignments are due at the start of class and must be easy to read.
Late assignments will not be accepted.
These questions and papers will be graded on the
following scale
Question
(out of 2)
0 – missing question
1 – question attempted, but
incomplete work
2 – question addressed
seriously
and in depth
In order to provide you with extensive comments,
there may be delays in returning these papers.
Exams
Two in-class exams will be given. Their focus is
largely conceptual and problem solving based. You should be able
to do and explain any of the assigned homework problems for the
material. You should be able to explain the concepts behind any
calculations, algorithms, etc. Material will come from lectures,
discussions in class, and the text. For example, you will need to be
able to explain clearly and with mathematical accuracy why we
can solve problems in certain ways or why various algorithms
or procedures work mathematically. You will also need to be able to use
and explain the use of the manipulatives relevant to the material.
Each exam will contain six questions: four of
the questions will be problems directly from the textbook. Two of
the questions will be more open ended and ask you to explain key
concepts from class. The exams will be graded as follows:
you will receive 40 points for attempting the exam. You may earn
up to 10 points on each of the questions. If you would like to
take the exam with less time constraints, you may choose to take it the
previous evening.
Make-ups for exams will be given only in extreme
cases with arrangements made with the instructor prior to
the exam or if there is a verifiable medical excuse or permission from
the Dean of Students. If you miss an exam and we have not made
arrangements prior to the missed exam, you must contact me before the
next class.
Final Project
This project will be a collection of
weekly question items that you will write up throughout the semester.
This collection
could one day be included in your professional portfolio to demonstrate
your level of mathematical understanding and preparation and your
ability to communicate mathematics in a clear and correct manner.
Details on this final project will be given out in class.
Reflection Logs
Your reflection log is a record of your thinking and
reactions to components of the course. These logs will be used
for various purposes including asking you to reflect on
challenges in
assignments or exams and asking you to reflect and react to
readings. Each courseday you will add an entry to your
log. Probably it should be at least a paragraph per day.
Each day will be out of 2 points. If you address the topic of the
day substantially you will receive 2 points. If you address it
less substantially you will receive 1 point.
Basic Skills Checks
As stated above, the goal of the course is a deeper
understanding of the content of the elementary school curriculum. At
the same time, there is a need to make sure that you can all do
the computations that you could one day teach. Therefore, throughout
the semester you will be given very short arithmetic quizzes which I
have called Basic Skills Checks. These quizzes will check your
computational competency (no calculators). They will be given prior to
each unit. If you do not pass each skills check (by demonstrating the
correct method for each question), your final course grade will be
lowered by one half letter for
each incomplete check. There will be the opportunity outside of class
to
retest in the event that you do not pass the skills check given in
class. course
page on a link entitled anonymous
feedback. Of
course, you are always welcome to approach me outside of class to
discuss these issues as well.
Math
Learning Center
This center is located in South Hall
332 and is open during the day and some evenings. Hours for the center
will be announced in class. The Math Learning Center provides free
tutoring on a walk-in basisQuizzes and Exams will be done individually unless otherwise directed of observance of religious holidays the
opportunity to make up missed work. You are responsible for
notifying
me no later than September 7 of plans to observe a holiday.
Postscript
This is a course in the mathematics
department. This is your mathematics content course. In
this course, you will develop a mathematical background necessary in
order to teach elementary school students. You will deepen your
understanding of gradeschool mathematics topics and connections.
You will not be learning how to teach mathematics to children, that is
the purpose of
your methods course in the school of education. As a
mathematician, I am trained to teach you mathematics, and I will do
that. I am not trained to teach you how to educate, and that is
not the goal of this course. Please keep this in mind.
We will be undertaking a great
amount of interactive group work in this course. You may
view these as games. If you come in eager to play, then you
will be more likely to be successful and perhaps occasionally enjoy
the games. If you come in saying "I don't want to play this
stupid
game," we will all be annoyed and frustrated, and the course as a whole
will be less successful. Please play nicely.
Out of necessity, I am more formal in class and more
personal out of class. If you ever want
additional help, please come to see me either during my office hours,
at an appointed time, or by just stopping by (I am frequently in my
office aside from the times that I will certainly be there). It
is important that you seek help when you start needing it, rather than
when you have reached desperation. Please be responsible.
Teaching is one profession where you have direct
impact on hundreds of lives. It is truly
an incredible responsibility. It is vitally important that
teachers set high expectations for themselves and their students.
Daily preparation of interesting, instructive lessons for twenty-five
or more active children of varying aptitudes is extremely
challenging.
I am dedicated to helping you prepare for this exciting career, and
will try to help you reach your full potential. Best wishes for a
challenging and fulfilling semester.
Schedule (This schedule is subject to change, but I
hope to hold mostly to this outline.) Two numbers separated by a
period refer to explorations that we will be studying that day in
class.
August 27 Introduction
29
1.1
31
1.4, Reflection Logs due
September 5 1.3 discussion, 2.1
7
History of number systems:
read text §2.3. Present pp. 103-110, Problem Set 1 due |
Welcome to the Math Success Center at Santa Ana College School of Continuing Education!
ABE Math
Page Content
COURSE DESCRIPTION:
Instruction in the four fundamental arithmetical operations of addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals. Provides learning activities which allow for remediation and mastery of the basic math skills. |
For a sophomore-level course in Linear Algebra. Based on the recommendations of the Linear Algebra Curriculum Study Group, this introduction to linear algebra offers a matrix-oriented approach with more emphasis on problem solving and applications. Throughout the text, use of technology is ...
For introductory sophomore-level courses in Linear Algebra or Matrix Theory. This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but... |
Using AIMMS to Teach Modeling and Operations Research
AIMMS is an excellent tool to support Modeling and Operations Research courses. The students can use AIMMS to implement assignments without the need to master a programming language: the understanding of the basic concepts like sets, parameters, variables and constraints is all they need to be able to build mathematical programming models and solve them with the available solvers, such as XA for linear and mixed integer programming, CONOPT for nonlinear programming, and AOA for mixed integer nonlinear programming. AIMMS' integrated visualization of results facilitates learning and interactive model development. Students generally find AIMMS easy and fun to work with!
Teaching Materials
Self-guided Tutorials
AIMMS comes with a Tutorial for Beginners, ideal for students; which includes all the necessary steps to build a model and a graphical user interface (GUI) in AIMMS. The tutorial can be completed in one hour and is currently available in English, Spanish and Hungarian. This tutorial covers all the information students typically need to start implementing assignments in AIMMS.
AIMMS also contains a Tutorial for Professionals, which covers more advanced material, such as rolling-horizon planning, database links and building an elaborate user interface. This tutorial takes about two days to complete. And finally, for a quick visual introduction to AIMMS, you can find a set of AIMMS videos on this website.
Teaching Book
AIMMS comes with a book "AIMMS Optimization Modeling", which contains optimization theory and a range of application examples. This can be an excellent addition to your own teaching material. A PDF version is available within each AIMMS installation and online, on our website. Hardcopies can be ordered from on-line bookstores.
G-AIMMS
In the G-AIMMS section on our website you find an AIMMS modeling game. When you have some basic modeling knowledge you can learn AIMMS and increase your modeling skills by playing the game.
Worked Examples
AIMMS contains a diverse set of worked examples, of applications in many varied fields. The applications described in the book "AIMMS Optimization Modeling" are also available as worked application examples within AIMMS. |
MATH 1240 Applied Mathematics
Lecture/Lab/Credit Hours 4.5 - 0 - 4.5
This course covers the development and application of the mathematical skills needed to solve problems related to industrial occupations. Topics include applications of arithmetic skills, measurement, and elementary algebra, geometry, and trigonometry. NOTE: MATH 1220 and MATH 1240 do not require MATH 0930, 0931, or 0960 as a prerequisite; however, MATH 0910 skills are necessary. MATH 1220 and MATH 1240 satisfy the math requirements in certain programs only. Check to see what the program advises to fulfill the general education math requirement. In most cases, these courses do not transfer to other institutions as math credit.
Prerequisites
(1) Within two years prior to beginning the course, either successful completion of MATH 0910 with a grade of P, or MCC placement test |
Brand New!
Who doesn't need help with math? Practically everybody… And we have the answer! Basic Shop Math is designed to assist students who need extra help or simply need to refresh their basic math skills. A practical, crystal-clear, Auto Shop centered math course that makes math understandable!
Most instructors start the school year with their first priority, shop safety. Right after that, math is often the next topic.
Basic Shop Math is designed to assist students with the basic math skills most often used in the Auto shop. Whether they need a little extra help or just need to refresh their math skills, this course covers what they need to know to work successfully in the Auto Shop. All math examples are expressed in terms associated with typical auto shop tasks or jobs.
Students who possess a solid understanding of fundamental math skills are more likely to succeed in the auto shop. |
Basic Algebra Shape-Up is interactive software that covers creating formulas; using ratios, proportions, and scale; working with integers, simple and multi-step equations. Students are able to track their
This bilingual problem-solving mathematics software allows you to work through 5018 Algebra equations with guided solutions, and encourages to learn through in-depth understanding of each solution step and...
This script defines the Matrix class, an implementation of a linear Algebra matrix. Arithmetic operations, trace, determinant, and minors are defined for it. This is a lightweight alternative to a numerical...
Stop searching through textbooks, old worksheets, and question databases. Infinite Pre-Algebra helps you create questions with the characteristics you want them to have. It enables you to automatically space...
Algebra Helper shows how signs change when moving a quantity from one side of an equation to the other. Algebra Helper is implemented in JavaScript. Currently works in Firefox and Internet Explorer 8....
Boolean Algebra assistant program
is an interactive program extremely easy to use.
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Interactive College Algebra course designed to ensure engaging, self-paced, and self-controlled e-learning process and help students to excel in their classes. Java- and web-based math course includes...
Mavscript allows the user to do calculations in a text document. Plain text and OpenOffice Writer files (odt,sxw) are supported. The calculation is done by the Algebra system Yacas or by the Java interpreter...
MathProf is an easy to use mathematics program within approximately 180 subroutines. MathProf can display mathematical correlations in a very clear and simple way. The program covers the areas Analysis,...
Calculator is a minimalist, easy to use Windows calculator that takes convenient advantage of the number key pad in your keyboard. Calculator can compute any Algebra expression instantly.
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The Algebra Coach is a program that explains Algebra step-by-step exactly the way your teacher does. It can do very complicated Algebra but is still easy to understand and use. The Algebra Coach can solve......
Infinite Pre-Algebra is a great tool for math teachers that allow them to generate tests and worksheets. Infinite Pre-Algebra covers all typical Pre-Algebra material, over 90 topics in all, from arithmetic...
Infinite Algebra by Kuta Software is a great application that can be really useful for all mathematics teachers. It was developed for quick creating of math quizzes to perform tests. The application contains...
Speedstudy Pre Algebra improve grades and test scores. Multimedia learning system makes even the toughest math concepts come alive. Great for new learners or students studying for college entrance exams... |
Many financial problems can be concisely expressed as matrices
(this is the proper plural form of matrix). What is a
matrix? Basically, it's just a rectangular set of numbers.
We classify matrices based on the number of rows and columns. A
matrix with 3 rows and 2 columns, for example, is called a 3 × 2
matrix. Is example is shown below.
It is common to refer to a matrix with a capital letter. For
example, let us call the matrix above A. Then the individual
elements (numbers in the matrix) are referred to with lowercase
letters (usually italicized). The element in row i and
column j is referred to as aij.
For example, the number "6" is in the 3rd
row and the 2nd column, so we say that a32
= 6. You may see this written as a[3][2] or a[3,2] if you read
text about matrices in plain text format, because subscripts are
not supported.
Matrices with the same dimensions can be added simply by
adding elements that are in the same positions. An example is
given below.
Similarly, matrix subtraction is a simple process. Simply
subtract numbers that are in the same positions.
Scalar multiplication is also rather simple with
matrices. A scalar is just a normal number, like
"5" or "-2.4." To multiply a matrix by a
scalar, simply multiply each element of the matrix by that
scalar.
The zero matrix is a matrix composed of all zeroes.
There is one zero matrix for each size of a matrix (for example,
there is a 3 × 4 zero matrix, a 5 × 5 zero matrix, etc.).
(The more mathematically inclined readers may note that the
above definitions show that the set of all m × n
matrices is a vector space.)
A square matrix is a matrix that has the same number of
rows and columns. The matrix in the first example above is not a
square matrix; the matrices in the second equation are square.
At this point you may be saying, "that's all very
nice, but what good is it?" The answer is that matrices can
be very useful, but first you need to know more about matrix multiplication, systems of
equations, and row operations. Matrix multiplication is not as
simple and intuitive as matrix addition. (For example, you can
multiply matrices that are not the same size!) We'll cover
it in the next section. Eventually we'll cover enough that
you'll be able to learn about the simplex
method, a powerful optimization method that can solve
problems with many variables.
One last note: matrices are always two-dimensional. There is
no "three-dimensional matrix," although the closely
related tensors can have multiple dimensions; they are
used by physicists, not by economists, so we won't cover
them. |
GCSE Modular Maths The Mathematics Department enter all pupils for a modular exam. This entails hard work during the 2-year course. There are three levels of entry: Foundation - Grades available D -... |
In math we often encounter certain elementary functions. These elementary functions include rational functions, exponential functions, basic polynomials, absolute values and the square root function.
It is important to recognize the graph of elementary functions, and to be ablo to graph them ourselves. This will be specially useful when doing transformations. |
11.1 From Arithmetic to Algebra 11.2 Evaluating Algebraic Expressions 11.3 Adding and Subtracting Algebraic Expressions 11.4 Using the Addition Property to Solve an Equation 11.5 Using the Multiplication Property to Solve an Equation 11.6 Combining the Properties to Solve Equations
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I am a visual learner so it would be easier to have more pics. like the one on pg. 110. More animations and graphs. I am also a global learner so more step-by-step explanations and more time on certain problems would be nice. |
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Starting at $49To facilitate those instructors that use a more hands-on approach to teaching, an extensive Exploration Activities manual accompanies the text. The manual contains a variety of explorations for each section, which are referenced in the text by an icon in the margin. Some explorations deal directly with the content of the chapter, often making use of relevant manipulatives or other hands-on activities. Other explorations extend the content of the section either mathematically or by building a connection to the K - 8 classrooms. Most of the explorations can be done individually or with groups and should take about 30 - 45 minutes to complete. |
Archive for the 'Science' Category
It is certain that university students who are majoring in science will get a lot of assignments besides having some direct experiments in the laboratory. Almost all of the assignments are hard enough to do. For example students who are majoring in chemistry will get complicated assignments. Tutorvista.com is one of the solutions for your academic problems. The website provides all kinds of chemistry help.
It covers the practical matters, like providing some tips for the laboratory activities, and the matters that are related to writing some papers, such as giving chemistry homework help. All chemistry branches are covered. So for you those who take organic chemistry for their concentration, you are able to get organic chemistry help. There is no need to struggle in making your homework because the website is able to give you online guidance. The guidance will be done by professional tutors who have enough experience in this matter. If you need more than a mere homework help that answers all of your questions related to your homework, you can also connect with the tutors directly to have some brief and detail consultation about your academic problems RSFYJQ489U43
The solution to a mathematical equation can be feasible or infeasible. There can be mathematical models with boundaries as well. Such models are termed as optimization models where the solution resides within a set of values. Usually such models are expressed with a set of constraints. For example, the classical functions of pricing the supply and demand for products, both these functions together create a fixed value for the price.
Here is a sample mathematical model
Objective: Maximize Profits from selling two products P1 and P2 at Price $3 and $4 respectively.
If you carefully notice the system of equations above, the first equation constantly increases for any value of P1 and P2. But the increase is restricted by equation number 2 which enforces a boundary on the system. Hence the solution set returns feasible values.
Equations can be both deterministic as well as stochastic. Stochastic systems are systems that do not have fixed values such as USD 3 as cost of product or labor hours as 2 hours per product. The expected values can be specified as a probability distribution. An Example of a probability distribution is the arrival rate of automobiles in a junction. One cannot determine the exact rate as the source would be dependant upon a lot of factors.
Simulation is an extended technique of analyzing variations of input and output using expected values for a large number of trials. Many contrasting system conditions can be specified and the simulation can be run for a large number of trials.
Math models are common place and are used to describe physical phenomena, astronomical phenomena and population growth. They are also used in production planning, manufacturing etc. In synopsis a mathematical model can create unbounded values or bounded values. A system with boundaries can be used to study extreme objectives such as profit maximization, time minimization etc.,
Hal Anger, a nuclear scientist, was the man who invented the gamma camera. Is a medical device, a camera, that is used to look at the internal organs and how they are functioning at the time. He invented it back in the 1950′s and it is still around today, nearly 70 years later.
The gamma camera works by through gamma rays. The patient is given a solution to drink or injected with a solution that contains a low level of radiation that will emit gamma rays. Depending on the organ that is being target to be looked at, a different type of solution is given. The reason for this is because certain types and mixes of the solution will be absorbed by different organs.
Once the solution is in the patient, they are then put under the gamma camera. The first part of the camera to receive the gamma rays coming from the patient is a part called a collimator. Like a colander that strains liquid from the solid, the nuclear medicine collimator on the gamma camera channels all of the gamma rays into the camera head that is full of crystals. What happens from there and how that becomes an image that you can see and make a diagnosis from is some complicated nuclear science stuff that most people do not understand anyways.
All that really matters for most of us is that it works. In particular, because since its invention, the largest success story of the gamma camera is that of the fight against breast cancer. It has been able to detect tumors much sooner, making the odds of survival in breast cancer patients much higher. quantity |
Algebra continues to be a major obstacle for students. Many stumble over the beginnings of the subject, and even those who fare well in two years of high school algebra are unable to see its power and utility in mathematics and related fields. Part of the reason for these difficulties comes from the disconnect between school algebra and algebra as a scientific discipline. Drawing on examples from Common Core, the monograph "Reasoning and Sense Making in Algebra," the recently released PARCC "Content Frameworks,'' and a four-year high school curriculum that my colleagues and I have developed, I'll talk about some ideas for making algebra more meaningful, coherent, and tractable for students and their teachers. |
Article Summary: Ninth grade math concepts can cover
several topics. However ninth grade math typically focuses on general math,
algebra, or geometry. This is due to whatever course a ninth grade student
is enrolled; there is no standard of what math subject should be taught
in the ninth grade.
What Math Concepts Are Taught in Ninth Grade?
Ninth grade math concepts can cover several topics. However ninth
grade math typically focuses on general math, algebra, or geometry.
This is due to whatever course a ninth grade student is enrolled;
there is no standard of what math subject should be taught in the
ninth grade. These are the math concepts that students should understand
by the end of the ninth grade based their course they are enrolled
and the National Mathematics Standards.
Numbers and Operations concepts focus on rational and irrational
numbers to solve complex mathematical problems. They also use the
quadratic equations to solve real world problems, providing sound
solutions. They develop an understanding vectors and real number systems
to solve problems. They use vectors and matrices to explain the properties
and representations that involve basic math computations through the
use of fractions, percents, decimals, and whole numbers. They also
learn to apply these concepts using mental math and pencil - paper
solutions.
Geometry concepts for ninth grade students' focus on developing
an understanding of two and three dimensional objects, such as: prisms,
pyramids, cubes, cones, spheres, cylinders, etc. They explore relationships
of shapes including congruence and similarity to solve the geometric
problems. They also use Cartesian coordinates to find the relationships
of two and three dimensional objects.
Ninth grade students use and understand geometric translations, reflections,
rotations, symmetry, and dilations of objects through the use of sketches
and matrices. Students construct two and three dimensional objects
using various tools to explain geometric situations. They use drawings,
models, and graphs to make connections of geometric situations in
other subject content areas and real world objects.
Algebra concepts students study are the relationships to functions
such as rates of change. In addition, they learn to use the multiple
variables for intercepts, zeros, and behaviors. They learn to use
more commonly used algebraic functions through the use of technology,
such as: graphing calculators and computer programs to solve and represent
equations.
Students learn to write and solve equations and inequalities using
mental processes, along with traditional paper and pencil. Students'
studies include the applications of manipulation of equations which
are logical and symbolic. They draw reasonable conclusions based on
their solutions, along with making connections with other mathematical
concept areas.
Measurement concepts focus on the use of standard and non-standard
(customary) units of measurement to determine the relationships between
different objects. This is also connected with geometry as they learn
how to measure the area, volume, and mass of different geometric shapes
using various tools. They learn how to measure all aspects of spheres,
prisms, pyramids, etc.. They use measurements to make drawings and
models of equations to explain a solution to a problem in both mathematical
and real world terms.
Data Analysis and Probability, students use appropriate language
to explain their findings in experiments and simulations. They learn
how to develop questions to help them find the differences between
several samples in a population. They develop studies of situations
to include the role of random and experimental surveys. They learn
to use and explain the univariate and bivariate in measurement and
categorical data. This information is used to develop scatter plots,
regression coefficients, and regression equations using technological
tools.
Students also study the application of sample statistics for developing
explanations using appropriate data analysis. This is used to develop
patterns of randomness for the probability that certain events may
be independent of other events. They learn to use simulations to explain
randomness of events.
Problem Solving for ninth grade students focuses on the development
of problem solving strategies to help them develop a fundamental understanding
of mathematics. Students use word problems and other real world simulations
in problems solving situations.
Representation concepts focus on students learning to collect
and organize data, then using the data to solve problems. Answers
are presented as models that are numerical, written, physical, and
social. They are able to draw graphs, charts, tables, and other forms
to explain how they solved a problem.
Connection concepts are designed for ninth grade students
to demonstrate how to make connections to real world applications
and other subject content areas. This includes making connections
with other concepts in mathematics.
Communicate their mathematics ideas in the form of sentences,
drawings, posters, and multimedia applications is another concept
that students need to master. This is used to ascertain their level
of understanding as they explain mathematical concepts to other students
and teachers.
Reasoning and Proof concepts are used to explain mathematical
findings and problem solving techniques. This is necessary so that
they develop skills on how to present logical arguments to math situations.
All of these mathematical concepts are used to develop a well rounded
base knowledge of mathematical ideas and language as students' progress
to higher levels of mathematics. |
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0764103032
9780764103032 more than 1,000 words, and directs readers to the page where the word is defined. Where needed, the definition is accompanied by examples. The book also features helpful illustrative diagrams--or instance, a full page demonstrating the geometry of the circle, another page showing quadrilateral geometric shapes, and still others showing ways of charting statistics, measuring vectors, and more. Here is an imaginative new approach to mathematics, a great classroom supplement, a useful homework helper for middle school and high school students, and a reference book that belongs in every school library. «Show less... Show more»
Rent Barron's Mathematics Study Dictionary today, or search our site for other Tapson |
0521398576
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The Mathematics of Projectiles in Sport: The mathematical theory underlying many sporting activities is of considerable interest to both applied mathematicians and sporting enthusiasts alike. Here Professor de Mestre presents a rigorous account of the techniques applied to the motion of projectiles. This equips the reader for the final section of the book in which an enlightening collection of sporting applications is considered, ranging from the high jump to frisbees and soccer to table tennis. The presentation should be accessible to most undergraduate science students and provides an ideal setting for the development of mathematical modeling techniques. «Show less
The Mathematics of Projectiles in Sport: The mathematical theory underlying many sporting activities is of considerable interest to both applied mathematicians and sporting enthusiasts alike. Here Professor de Mestre presents a rigorous account of the techniques applied to the motion... Show more»
Rent The Mathematics of Projectiles in Sport today, or search our site for other Mestre |
UMS is designed to solve and explain mathematical problems entered by the user. This program provides step by step solutions to most problems in arithmetic and algebra for middle- school students. All solutions are accompanied by step by step verbal and written commentaries. UMS Basic can completely solve all problems drawn from the following domains of mathematics: • arithmetic operations with common, decimal and mixed fractions • finding one factor of polynomial • complete factoring of polynomial • simplification of numerical expressions, including expressions with radicals • simplification of rational algebraic expressions • expansion of rational algebraic expressions • rational equations, where the equation can be with absolute values and with parameters • simultaneous rational equations, where each equation can be with absolute values and with parameters • rational inequalities, where an inequality can be with absolute values • systems and sums of rational inequalities, where each inequality can be with absolute values |
Book Description: Tom Pirnot believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking behind the subject matter, so that students are able to truly understand the material and apply it to their lives. This textbook maintains a conversational tone throughout and focuses on motivating students and the mathematics through current applications. Ultimately, students who use this book will become more educated consumers of the vast amount of technical and mathematical information that they encounter daily, transforming them into mathematically aware citizens.
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Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it |
This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and algebraic properties. New concepts include solving two-step equations and inequalities, graphing linear equations, simplifying algebraic expressions with exponents, i.e. monomials and polynomials, factoring, solving systems of equations, and using matrices to organize and interpret data.
Students will be actively engaged using concrete and virtual materials and appropriate technologies such as graphing calculators and computer software. However, facility in the use of technology shall not be regarded as a substitute for a student's understanding of quantitative concepts and relationships or for proficiency in basic computations. Students will also identify and apply real-life applications of the mathematical principles they are learning to science and other disciplines they are studying.
Mathematics has its own language, and the acquisition of specialized vocabulary and language patterns is crucial to a student's understanding and appreciation of the subject. Students are encouraged to use correctly the concepts, skills, symbols, and vocabulary identified in the course.
Problem solving is integrated throughout the different components of the curriculum. The development of problem-solving skills is a major goal of the mathematics program. Instruction in the process of problem solving will help students develop a wide range of skills and strategies for solving a variety of problem types.
Here is a description of what you will find in the navigation bar at the top.
Modules
Click on the module to see all of the individual lesson topics or select a module topic from the list.
SOL
View the Algebra 1 SOLs, Review Materials, Released Tests, and a link to the Department of Education web site.
Resources
Additional materials that are beneficial to students and teachers that are not included in the Modules or Lessons. |
In this textbook, the foundations of mathematics are made explicit and the reader is guided through the background knowledge and understanding that are required for the subject, offering a well-structured overview of the important issues to be considered when learning about mathematics on a Primary QTS course, and a coherent approach to the content to be found in the standards for QTS, the National Curriculum at Key Stages 1 and 2 and the numeracy strategy.
The authors aim to help teachers review and restructure the understanding of mathematics gained during their education, progressing from partial memories of a few process to an understanding of why the skills they were taught make sense and how they fit into a coherent mathematics curriculum, arguing that to teach mathematics effectively it is not enough to be able to do the mathematics, you need to understand why you do what you do. Aimed at all teachers of primary mathematics, this book is also likely to be valuable to secondary teachers, parents, administrators and others interested in the foundations of school mathematics.
Written for trainee and practicing teachers, this book de-mystifies the primary mathematics UK curriculum and offers a valuable reference for effective mathematicss teaching. |
mathematicsUsing Maple, available from Adept Scientific (Letchworth, Herts), instructors can easily create interactive mathematical applications that help their colleagues and students solve problems, visualise solutions and explore conceptsAll Press ReleasesMathematics and Music: Composition, Perception, and Performance explores the many links between mathematics and different genres of music, deepening students' understanding of music through mathematics.
Using Maple, available from Adept Scientific (Letchworth, Herts), instructors can easily create interactive mathematical applications that help their colleagues and students solve problems, visualise solutions and explore concepts.
Lanika announces Maplesoft's major new initiative to support teaching and learning. The Möbius Project makes it easy to create rich, interactive mathematical applications, share them with everyone, and grade them to assess understandingFor the third consecutive year, President Barack Obama used his State of the Union address to call for increased investment in science and technology. More than 40,000 Wisconsin students heed this call each day.
The Hong Kong Polytechnic University (PolyU) signed today (22 January) an agreement with the Australian Mathematics Trust (AMT) to launch the "Mathematics Challenge for Young Australians" (MCYA) in Hong KongThrough a careful treatment of number theory and geometry, Number, Shape, and Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofsMyAcademicWorkshop™ is a differientated online assesment system that helps place students into appropriate mathematics courses and helps increase student engagement and sense of accomplishment for students struggling in mathematics.
This book presents advanced mathematical descriptions and methods to help readers achieve more thorough results under more general conditions than what has been possible with previous results in the existing literature.
Physical Oceanography: A Mathematical Introduction with MATLAB® demonstrates how to use the basic tenets of multivariate calculus to derive the governing equations of fluid dynamics in a rotating frameCarnegie Mellon Hosts 30th Annual Charles Drew Science Fair on March 10, 2012. The fair was created by Adolphus Patterson and C. Richard Gilcrese to encourage students to take an interest in careers in science, technology, enigneering, & math (STEM)."We're looking forward to having Big Ideas Learning join The Balancing Act on Lifetime TV. The Balancing Act Lifetime viewers will have a chance to learn how to help their children master mathematical benchmarks at each level."
Students do not need more STEM- Science, Technology, Engineering, and Mathematics -, they need immersion in the invention/innovation design process. Support Fluke - the wealth building game of accidental inventions.
Preparing the UK's next generation of engineers and technicians has taken a leap forward with the official opening of a state-of-the-art technology centre at leading independent training provider, S&B Automotive Academy in Bristol, UK. |
freshman-level prealgebra courses. This text's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students. The primary focus of the pedagogy, presentation and other elements is to ease the transition into algebra; for example, emphasis is placed on basic arithmetic operations within algebraic contexts. The Second Edition includes a greater integration of NCTM and AMATYC standards, including more emphasis on visualization, problem solving and data analysis. ... MOREtyle="width:100%" />Elayn Martin-Gay believes "every" student can succeed and that is the motivating force behind her best-selling texts and acclaimed video program. With Martin-Gay you get 100% consistency in voice from text to video! "Prealgebra 5e "is |
If learners in the classroom are to be excited by mathematics, teachers need to be both well informed about current initiatives and able to see how what is expected of them can be translated into rich and stimulating classroom strategies.
The book examines current initiatives that affect teaching mathematics and identifies pointers for action inGraduate textbooks often have a rather daunting heft. So it's pleasant for a text intended for first-year graduate students to be concise, and brief enough that at the end of a course nearly the entire text will have been covered. This book manages that feat, entirely without sacrificing any materia more...
The leading reference on probabilistic methods in combinatorics-now expanded and updated When it was first published in 1991, The Probabilistic Method became instantly the standard reference on one of the most powerful and widely used tools in combinatorics. Still without competition nearly a decade later, this new edition brings you up to speed on... more...
In 2003 the British Combinatorial Conference conference was held at the University of Wales, Bangor. The papers contained here are high quality surveys contributed by the invited speakers, covering topics of significant interest. Ideal for established researchers and graduate students who will find much here to inspire future workA unique approach illustrating discrete distribution theory through combinatorial methods This book provides a unique approach by presenting combinatorial methods in tandem with discrete distribution theory. This method, particular to discreteness, allows readers to gain a deeper understanding of theory by using applications to solve problems. The... more... |
Pre-Algebra
Pre-Algebra concepts
are presented in this unit, including order of operations, and
writing algebraic expressions and equations. Try our Pre-Algebra lessons below,
or browse other units of instruction. |
The purpose of the course is to introduce mathematical modeling, i.e., the construction of mathematical structures which capture relevant physical phenomena. The course will systematically explore mathematical ideas and tools used to study the natural world. Particular emphasis will be placed on the process of creating a mathematical model starting from a physical scenario. Typically this process will begin with an experiment either demonstrated in the W. G. Pritchard Lab or performed by the students in class.
Once a particular model has been developed, students will use mathematical analysis and experimentation to determine the properties and relevance of the model, and to make predictions. Often the model can be satisfactory; however, many times one also finds new features of the system that are not adequately accounted for in the model, and the process begins again. It is this cycle the course will focus on. For a given phenomenon (e.g., flow of viscous fluid, pendulum motion) several models may be compared and contrasted, and possible simplifications will be discussed.
A significant aspect of the course is its laboratory component, in which the students will perform experiments or observe demonstrations. However, the main emphasis will be placed on creating and rigorously analyzing the mathematical aspects of the models. Instead of presenting a finely tuned model for a given phenomenon, this course will try to convey some of the heuristic, intuitive, and mathematical ideas employed in modeling.
Examples of physical systems to be considered include: simple and compound pendulum motion, chemical oscillations, water waves, and elastic behavior of polymer solutions.
The course is open to a wide range of undergraduate as well as graduate students with majors in mathematics, biology, chemistry, engineering, and physics. The course should be accessible to students with some basic knowledge of mathematical analysis and differential equations. Main topics include: modeling with ordinary differential equations; bifurcation theory and stability; traveling waves in epidemics, chemical reactions, free fluid surfaces, and polymer solutions; fluctuations in nature, stochastic differential equations and chaos |
Placement Information
Department of Mathematics & Statistics
Placement The main objectives for advising students taking math courses are.
determine the developmental math status of the student
determine the level of courses they are prepared to take
determine which courses they should take
help students get registered
Tools to achieve these objectives are can be found in this simplified placement grid and an outline on the topics of the local placement exam. Placement for higher level courses is also available. See below for details.
Developmental Math Status The Texas Success Initiative requires the university to provide mathematical development for some students. Students identified in this way are called Developmental Math Students. All other students are College Level Math students. Developmental Math Students are those who have not yet scored at least 230 points on the TASP or THEA exams, or are placed in MATH 0399 or MATH 0398 by the local placement process (described below). Students move from Developmental Math status by scoring at least 206 on the THEA exam and completing MATH 0399 or a higher level course under a Developmental Math Success Contract. Such students need to have their math registration supervised by the TLC. College Level Math students are all others. They should be taking math courses that satisfy their core curriculum requirement, their degree requirements or electives.
Levels of MATH Courses Introductory MATH courses are offered at the following levels. Courses at a given level are prerequisites for the courses at the next level.
Level 1. MATH 0398, Introductory Algebra
Level 2. MATH 0399, Intermediate Algebra
Level 3. MATH 1314, College Algebra
Level 4. MATH 1316, 1324, 1325, 1342, 1442, 1470, 2312, 2342
Level 5. MATH 2413, MATH 2305
Consult the catalog for prerequisites for higher levels of math courses.
The Placement Process Placement in MATH courses uses the following criteria. Tables are printed below to give details of how the criteria are used. The overall placement is the highest resulting placement from the individual criteria.
TASP/THEA Exam Scores
SAT Quantitative Score
ACT Math Score
Precalculus in High School with a grade of C (70) or higher
Previous College/University Credit in MATH courses
Local Placement Test
Placement from External Test Scores
For TASP/THEA scores alone
Less than 206: Level 1
Between 206 and 259: Level 2
At least 260: Level 3
SAT/ACT Scores with Precalculus in High School
SAT at least 620, ACT at least 27: Level 5
Otherwise: Level 4
SAT/ACT scores without Precalculus in High School
SAT at least 380, ACT at least 16: Level 2
SAT at least 450, ACT at least 18: Level 3
SAT at least 500, ACT at least 21: Level 4
SAT at least 620, ACT at least 27: Level 5
Placement by previous College/University Course Credit with a grade of C for a course taken at TAMUCC or accepted in transfer at a given level serves as prerequisite for courses at the next higher level. Note: Developmental Math courses (Levels 1, 2) are generally not posted on SIS but are accepted as prerequisites if a transcript is shown to the MATH program.
Local Placement Exam Actually, there are three separate placement exams. Most students take the Algebra Exam. Possible placements from that exam are Levels 1 through 4.
Students seeking placement into Level 5 or higher should take the 'Ready for Calculus' exam or the Calculus II exam.
The placement exams are normally offered Fridays at 10 AM, but other times are available by prior arrangement. Students should contact the Mathematics Program at 825-2459 to schedule taking the exam. Results are usually available that afternoon and are posted on Colleges under the Students: Student Information tabs and the Tests button.
Registration Prerequisites for MATH classes are as posted in the university catalog and are enforced through registration controls. Students incorrectly excluded from courses for which they have prerequisites should contact the Mathematics Program at 825-2459 and be prepared to show how they meet the prerequisites. Exceptions to Course enrollment caps require Math Coordinator and College of Science and Technology Dean approval. Forms requesting the exception are available from the College in FC 179 or from the department in CI 301. |
This course covers a wide variety of topics and is designed for students who have already taken algebra 2 and geometry. Students who have a serious interest in mathematics or who plan to major in a scientific field should take either the SL or HL course.
Areas of study include an introduction to naïve set theory, measurement, logic, a review of algebra, functions (including but not limited to topics such as domain, range, one to one, inverse functions, graphing and shifting graphs, the algebra of functions, and quadratic functions), analytic geometry, full coverage of trigonometry including the law of sines and law of cosines from first principles, sequences and series, financial mathematics, probability and statistics, and basic coverage of differential calculus.
TEXTBOOKS AND REQUIRED MATERIALS:
1)Mathematics for the International Student: Mathematical Studies SL. 2nd edition. Haese and Harris.
2) IB classes require a graphing calculator. Please consult the list of approved calculators before you make a purchase. I highly recommend the TI-84 Plus. Roughly speaking, calculators capable of symbolic mathematics are not acceptable.
Homework will be collected at the start of each class. Any student with incomplete homework will be assigned to tutorial to make it up. Quizzes will be announced at least three days in advance and must be made up on the first day back in case of absence. Each semester comprises two quarters and the semester grade will be the average of the two quarterly grades. |
Calculus: An Active Approach with Projects
Overall, I heartily recommend ... Calculus: An Active Approach with Projects. Though I have not yet experimented with such a broad implementation of activities and projects in my calculus classroom, Calculus: An Active Approach with Projects certainly provides concrete tools, helpful suggestions and the inspiration to try. — Susan D'Agostino, MAA Online
This volume contains student and instructor material for the delivery of a two-semester calculus sequence at the undergraduate level. It can be used in conjunction with any textbook. It was written with the view that students who are actively involved inside and outside the classroom are more likely to succeed, develop deeper conceptual understanding, and retain knowledge, than students who are passive recipients of information.
Calculus: An Active Approach with Projects contains two main student sections. The first contains activities usually done in class, individually or in groups. Many of the activities allow students to participate in the development of central calculus ideas. The second section contains longer projects where students work in groups outside the classroom. These projects may involve material already presented, motivate concepts, or introduce supplementary topics.
In addition to facilitating active student learning, the material will foster student comprehension of calculus as a unified subject. It provides many opportunities for students to make connections between different calculus topics. Unifying threads appear throughout the activities and projects. These threads include graphical calculus, distance and velocity, multiple representations of functions, estimation and approximation, and mathematical modeling.
Student thinking and communication are promoted through use of activities and projects where students need to organize their thinking, determine problem-solving strategies, and clearly communicate results to others.
Instructor materials contained in the volume include comments and notes on each project and activity, guidelines on their implementation, and a sample curriculum which incorporates a collection of activities and projects.
Note: Teachers may copy or print out the activities or projects to use as handouts for their students.
Print-on-Demand (POD) books are not returnable because they are printed at your request. Damaged books will, of course, be replaced (customer support information is on your receipt). Please note that all Print-on-Demand books are paperbound. |
From the Publisher: Authors Wayne Winston and Munirpallam Venkataramanan emphasize model-formulation and model-building skills as well as interpretation of computer software output. Focusing on deterministic models, this book is designed for the first half of an operations research sequence. A subset of Winston's best-selling OPERATIONS RESEARCH, INTRODUCTION TO MATHEMATICAL PROGRAMMING offers self-contained chapters that make it flexible enough for one- or two-semester courses ranging from advanced beginning to intermediate in level. The book has a strong computer orientation and emphasizes model-formulation and model-building skills. Every topic includes a corresponding computer-based modeling and solution method and every chapter presents the software tools needed to solve realistic problems. LINDO, LINGO, and Premium Solver for Education software packages are available with the book.
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This introduction to aspects of semidefinite programming and its use
in approximation algorithms develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others.
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Stochastic programming the science that provides us with tools to
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Bringing a new vitality to college mathematics
What price are we willing to pay for 'correct answers'? What gains (benefits) should students expect for dealing with applications in a math class?
In our beginning algebra class this week, we spent much of our time on applications. Many of these were the typical puzzle problems involving tickets and cars, integers and angles. As is normal for this course, students really wanted some magic — a rule that would help them get the correct answer for all of the problems. Some of the students remembered some magic from a prior math class; one piece of magic was the word 'is' … the other piece of magic was a triangle (for mixtures).
We often provide rules (whether perfect or not) that are meant to help students get more correct answers for applications (broadly stated as word problems involving a context). We tell students that "of" means multiply, and that "is" means equal; the prototype for both rules is the "a is n% of b" template (a worthless model, as normally taught). Students who have experienced this 'correct answer' driven course encounter many problems when faced with a narrative about an application, where 'of' is the normal preposition and 'is' is the normal verb connecting phrases. We train our students to surface-process language for the sake of correct answers, and wonder why students continue to have problems with applications.
One of the most challenging problems we did this week was this simply-stated problem:
A store claims that they markup books by 30%, and the selling price for one book is $79.95. Find the cost of the book to the store (before the markup was added).
Every student in this particular class was a graduate of our pre-algebra course, where this same problem was done as part of a longer chapter on percents and applications. Every student in this class wanted to either multiply by 30% or divide by 30%; a few students thought that there was a second step where they needed to add or subtract this result.
Quite a few of the students could do this problem:
A store sells a book that has a cost of $61.50, and they have a markup on books of 30%. Find the selling price.
Their success on this arithmetic problem was not based on understanding the words any better (the words are the same). Their success was based on the 'magic' rules we had given them that happen to work: multiply by the percent, add or subtract if needed.
The whole point of experiencing applications such as these is to build up the student's mathematical reasoning. There might be magic in the world, but magic is not reasoning. Correct answers based on locally-working magic is worse than wrong answers based on weak reasoning. If our courses include applications, keep the magic of "is" out of the course … and all other magic.
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2 Comments
I graduated in 1980 with a Bachelor of Science Degree in Math. No matter which way the problem was stated, I could easily solve it. Are you saying that some students have difficulty understanding that the word "of" has more than one meaning, and that both meanings used in the same story problem confuse them? If that is the case, I think that those students will have difficulty in any area of life.
Yes, that is what I am saying … along with an observation that math teachers sometimes suggest rules which are only true in specific localities. My students have 'difficulty' in multiple areas, as you suggest. I am more concerned with how we (math instructors or professors) respond to student difficulty. |
Practical
Problems in Mathematics for Electronic
Technicians Practical,
Easy-to-Understand Problems Assist Students for The Practical
Problems that Professional Electronic Technicians Face Everyday
By Stephen Herman
- Sixth Edition
Success in the electronics field
requires a substantial background in mathematics. This updated book
is written to provide beginning students with these needed skills.
Practical, easy-to-understand
problems help prepare students for the types of problems that
professional electronic technicians face everyday. As part of the
successful Practical Problems in Mathematics series, this edition
features expanded coverage of scientific notation, increased
problems to be solved using a calculator, additional information on RLC circuits, and a new unit on simultaneous equations that includes
coverage of Kirchoff's Law. |
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Science does not exist in a vacuum and, therefore, shouldn't be taught that way. In that spirit, Activities Linking Science with Mathematics, K-4 is a hands-on guide for preservice and inservice elementary school teachers who want to connect science instruction with other areas of study including visual arts, social sciences, language arts, and especially math. The 20 discovery-based and academically rigorous activities provided in this volume enrich students awareness of the world around them, encourage their natural curiosity, and promote the development of their problem-solving skills.
Introductory Algebra prepares students for Intermediate Algebra by covering fundamental algebra concepts and key concepts needed for further studyWhether you are student who needs more practice than your textbook provides or a professional eager to brush up on your skills, 1001 Math Problems gives you all the practice you need to succeed. The ultimate learn-by-doing preparation guide, 1001 Math Problems will teach you how to: Prepare for important exams Develop multiple-choice test strategies Learn math rules and how to apply them to problems Overcome math anxiety through skills reinforcement and focused practice How does 1001 Math Problems build your math skills?
The book is the english translation of a former italian edition. Its aim is to provide students with the first mathematical tools regarding Linear Algabra. The challenge is to explain to all the first rudiments of a fundamental knowledge for science and technology. The text has been written by a mathematicians in order to meet the expectations of general public. |
Please note, this offer price only applies to individual customers when ordering direct from Oxford University Press, while stock lasts. No further discounts will apply. If you are a bookseller, please contact your OUP sales representative.
The Chemistry Maths Book
Broad coverage captures all the key mathematical concepts and theories with which a chemist should be familiar, making it the ideal reference throughout your studies.
The author's unfussy approach lets the subject speak for itself, using a combination of straightforward explanations and reinforcing examples.
The much-praised lucid writing style leads the student through even the most challenging topics in a steady, engaging way.
Extensive range of worked examples demonstrate every important concept and method in the text, to help the student grasp the material being presented.
End of chapter exercises encourage the student to learn through hands-on practice.
Online Resource Centre features additional resources for registered adopters of the text, to facilitate its use in teaching.
New to this edition
Contents reorganized to link the text and examples more closely with the exercises at the end of each chapter: it is now easier for the reader to try for themselves the exercises that are directly relevant to the topic they have been reading.
Material within some chapters has been reorganized to make the development of the subject more logical.
Extensive changes to chapter 1, retitled 'Numbers, Variables and Units' include a new section, Factorization, factors, and factorials, which fills a gap in the coverage of elementary topics and a rewritten and much enlarged section on units.
Chapter 2 has been revised to accommodate more discussion of the factorization and manipulation of algebraic expressions.
In chapter 9, the section on line integrals has been rewritten to clarify the relevance of line integrals to change of state in thermodynamics.
In chapter 13, a revision of the section on the Frobenius method, with new and more demanding examples and exercises.
In chapter 19, revised treatment of eigenvalues and eigenvectors, with new examples and exercises, to improve the flow and clarity of the discussion.
A full set of worked solutions to the end of chapter exercises is now available in the book's Online Resource Centre.
The Chemistry Maths Book provides a complete course companion suitable for students at all levels. All the most useful and important topics are covered, with numerous examples of applications in chemistry and the physical sciences.
Taking a clear, straightforward approach, the book develops ideas in a logical, coherent way, allowing students progressively to build a thorough working understanding of the subject.
Topics are organized into three parts: algebra, calculus, differential equations, and expansions in series; vectors, determinants and
matrices; and numerical analysis and statistics. The extensive use of examples illustrates every important concept and method in the text, and are used to demonstrate applications of the mathematics in chemistry and several basic concepts in physics. The exercises at the end of each chapter, are an essential element of the development of the subject, and have been designed to give students a working understanding of the material in the text.
Online Resource Centre: The Online Resource Centre features the following resources: - Figures from the book in electronic format, ready to download - Full worked solutions to all end of chapter exercises
Readership:
Students of chemistry at all levels.
Erich Steiner, Honorary University Fellow and former senior lecturer at the University of Exeter, UK
Review(s) from previous edition
"It seems well suited both for its stated purpose and as a "brush-up" book for undergraduates, graduate students, and others. The mathematics are carried out briskly and with very little dressing ... there is much material to cover here and it works well through Steiner's particularly lucid presentation. The notation is standard and clear ... I am impressed with this book, I am sure that it will remain open on my desk and will become well worn in short order.
- C. Michael McCallum, University of the Pacific, Journal of Chemical Education, Vol. 74 No. 12 December 1997 |
This book contains 25 full color transparencies, an activity worksheet for each transparency, and a teaching suggestions page for each transparency.
This book offers "a unique way to involve students in real life problem solving. Comprehensive Student Activity Masters and Teaching Suggestions provide a convenient way to connect mathematics with real-world situations."
Included for each chapter of the student text:
1) Two or more full color transparencies
2) Student Activity Master
3) Teaching Suggestions
This book is brand new and is in mint condition. The transparencies and worksheets are perforated and three-hole punched so that the pages can easily be removed and placed in a three ring binder. Comes from a smoke free home. |
Pages
Monday, July 11, 2011
How to put modeling at the center of NY State's Alg2/Trig
Q: What new skill should Algebra 2 students leave with?
A: The ability to model a system mathematically.
NY's A: Yeah, that. Also, how to solve an absolute value equation, how to employ Degree, Minute, Second notation to represent an angle, how to graph an inverse cosine curve...
And that, basically, is the challenge in reorienting the Alg2/Trig course around a single question or theme.
Still, I've been working on reorienting NY's Alg2 curriculum along these lines. This isn't exactly ground-breaking: Kirk Weiler's e-text, for instance, points at such an orientation. He starts by discussing functions, and then introduces different families of functions that end with a regression and modeling unit.
Here's what I would like to do differently in organizing the curriculum:
1. I want to bring the modeling and regression to the beginning of the function unit, to motivate our study of the function family.
2. I want to discuss a concrete example of a function, such as the familiar linear functions, before talking about functions in the abstract.
3. I want all the other stuff -- and boy oh boy is there a lot of other stuff -- to fit into the larger discussion about modeling.
1 and 2 are doable. 3 is hard. Still, there are some things that can be done to integrate the various skills of the course. For instance, much time is spent in Alg2 solving equations. By the end of the year, students should be able to solve absolute value equations, radical equations, quadratic equations, trig equations, exponential equations, log equations and rational equations.
These sort of skills, however, become necessary when you've mathematically modeled a system, found a representative function, and now wish to extrapolate. You're either going to be evaluating an expression, or solving an equation. If you think about the curriculum in this way, you have functions at the center of the curriculum, the functions are there for modeling, and a clear distinction between evaluating an expression and solving an equation will be constantly reinforced.
Ditto for inequalities.
So functions, modeling, and solving equations are taken care of. They fit into the larger framework. What's left over is all the stuff that has to do with manipulating expressions. For instance: simplifying radicals, simplifying complex exponents, simplifying complex fractions, exponent rules, etc. How do these things fit into the larger framework?
The best that I can do now is to say that these are upgrade packages, so to speak. The ability to manipulate expressions will allow us to have an easier time evaluating function expressions for a value, or expressing answers to function equations. So I think what I'm going to do is be explicit that these areas don't directly fit into our modeling narrative -- they're not used to describe or extrapolate based on data -- but they're excurses, upgrade packages that will allow us to model certain relationships more accurately.
In summary: 1) Bring statistics and regression to the foreground, to motivate the study of functions. 2) Put extrapolation at the center of function units. Extrapolation motivates both the evaluation of expressions and the solving of equations. 3) Explicitly bring out all the leftovers into upgrade packages, that will assist us in our next modeling exercise.
My next post will organize Alg2 standards into this framework. The post after that, hopefully, will reflect critically on this and think about what some of the problems of this will be. |
Click on the Google Preview image above to read some pages of this book!
Thisÿclassic undergraduate textÿelegantly acquaints students with theÿfundamental concepts and methods of mathematics. In addition to introducingÿmany noteworthy historical figuresÿfrom the 18th through the mid-20th centuries, it examinesÿthe axiomatic method, set theory, infinite sets, the linear continuum and the real number system, groups, intuitionism,ÿformal systems, mathematical logic, and other topics. |
Additional Notes on Algebra: Themes, Tools, Concepts
There are abundant teacher notes in ATTC's Teacher's Edition, by Anita Wah and Henri Picciotto. This page of additional notes will mostly be useful to you if you are planning on adapting or editing a lesson for use with your students. (There are links to a number of adapted / edited lessons here.)
What you will find below:
- Errata and formatting tweaks
- Additions and corrections to the answers given in the Teachers' Edition
- Other comments that have occurred to me in the intervening years
Your contributions to this page are welcome. If you find an error, or have an idea on how to improve a lesson, please e-mail me!
Many if not most lessons in this book should be started with the book closed. The explorations that initiate many of the lessons offer an opening that should make it much easier for students to understand what the questions addressed in the lesson are. After working on that, you can determine how much of the lesson students should do in the book.
Like many textbooks, ATTC contains a lot more material than can be covered in a single year. You need to be judicious in selecting which lessons to do, and which parts of each lesson.
While non-traditional, the sequencing of the book makes sense. Chapters 1-4 are essentially pre-algebra. Ideally, students would have already seen much of this. Chapters 5-9 are the core of an introduction to algebra. The final chapters offer additional topics to choose among, many of which I prefer to teach in later courses. However, the starts of chapters 8 and 13 should happen earlier than is suggested by their placement in the book.
A Word from the Authors
We listed all the math educators whose ideas influenced us in the writing of the book. Would that all math curriculum authors listed their inspirations!
Letters
You might create your own version of the 'Dear Student' letter.
You can ask your students to write a 'Dear Teacher' letter, in response to the 'Dear Student' letter. Ask them to include information about their math background and history, their math fears, their math hopes. This can be the start of a useful conversation.
Later, you can follow up by asking students to have their parents read 'Dear Parents'. Students can interview their parents about their algebra (or other math) memories. Ask students to compare their algebra experience with their parents'. (Perhaps in the form of a letter to you, or as a newspaper article about changes in math education.)
Lesson 1.1
For #17, one student came up with the generalization: 'subtract the shortest perimeter from the longest; divide by 2; add 1'
Lesson 1.2
#17: change to 'for an area of 20'
Lesson 1.4
Repeat the instructions from problem 1 in #2-4.
#5: Suggest writing algebraically before evaluating. Figure seems to consist of three parts.
1.A
Suitable for homework: 7
The table is difficult to read. It needs to be broken up into four separate tables, and an illustrated example is needed.
Lesson 1.5
#11: You must use more than one block.
Lesson 1.6
Suitable for homework: 1, 6, 10-22
Lesson 1.7
The subtitle and teacher notes for the last three problems correspond to #20-21 in 1.11.
#29-31 should be moved to be before #21, and retitled More Perimeter Puzzles.
Lesson 1.8
The Teachers' Guide says: 'It's preferable you give no help'. Actually if needed, use the following hints:
start by analyzing one-pane windows
don't forget that window panes include both glass and wood -- how much of each material may help determine the price
The windows should look (graphically) like the ones in Ch 7.2, and they should each be labeled, a, b, c, etc. to facilitate discussion.
1.B
Make clear that no window should be left uncovered.
#4: reword so that report is a letter to Ms Tall. (In fact the strategy of suggesting a specific audience for the reports should be used whenever possible.)
#3, 4 solution: 5 undecided windows can get long drapes, $17 left over
Lesson 1.11
In the instructions between #7 and #8, and beyond, use the word 'change' instead of the word 'step'.
#16-21 should be below the line, anywhere in the chapter.
#17 sols are missing. #19 sols need to be illustrated. We need a better explanation of why 27 is impossible.
#24: the reference is to the previous problem, not to problem 11.
Lesson 1.12
Suitable for homework: (using dot paper) 4-6, 11-17
Useful for assessment: 17
#1: 'at least three', rather than 'as many as you can'
#15a and c: Note that there are only four right and acute triangles that satisfy the given conditions. Accept solutions that are mirror images of each other, or seize the opportunity to show that there are obtuse triangles that work.
1. Essential Ideas
Instructions preceding #2-3 are unclear. Should be rewritten as a, b, c, d:
Lesson 3.1
Lesson 3.2
#11: Change to 'When x is negative, y is greater than 5. What does this say about two negatives?'
Delete #13b, which is very tedious.
Lesson 3.3
#18. Do not use a calculator.
Lesson 3.4
#1 is not an exploration. If this cannot be changed, at least the end-of-exploration graphic should be inserted after #1.
4a. 'numbers' instead of 'answers'
5, step (2): Multiply by 4.
A
Remove the thick horizontal line from the tables at the top of the page. (OK in the one at the bottom.)
Lesson 3.5
#23: change 'when' to 'whenever'
Lesson 3.6
#10: …in problem 9, if there is no remainder, write the related…
We need more problems of the type in #11. Not necessarily here.
#19: put a 'y2' to the left of the bottom row of the table
#18-19: no sols given!
Lesson 3.7
#4: put 'different' in italics
#12: not a report, but a key
#13: Hint: what is 1/82 as a decimal?
#17: not an exploration
Lesson 3.8
#6: 'approximately' should be in italics
#5: suggest a scale for the axes, either in the TG, or right in the problem. Perhaps a figure.
Add a problem after #15: redo #8, using the formula and trial and error
Lesson 3.9
#14 solution: 'beddy-bye' is in the Random House dictionaries.
Lesson 3.10
#1 should not be there. Kids resent making a diagram that's already in the book. (As a rule, problems where they are asked to make tables, diagrams, or graphs that are already in the book should be removed, and replaced with questions involving reading the table, diagram, or graph.) |
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