Technology & Engineering
Basic Algebra
Basic algebra is a branch of mathematics that deals with the manipulation of symbols and the rules of operations. It involves solving equations, simplifying expressions, and finding unknown variables. It is a fundamental tool in many fields, including technology and engineering.
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9 Key excerpts on "Basic Algebra"
eBook - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
Algebra and Basic Math 2 CONTENTS 2.1. Fundamentals Of Algebra ................................................................. 38 2.2. Operations On Monomials and Polynomials ..................................... 40 2.3. Linear Equations In One Variable ...................................................... 46 2.4. Problems To Solve ............................................................................. 58 References ............................................................................................... 64 Fundamentals of Advanced Mathematics 38 Mathematics is a vast subject, which includes arithmetic, calculations, statistics, and various other branches. One of the significant fields that make up the great subject of mathematics is ‘Algebra.’ Algebra is the basis for all the topics that come up in the advanced form of mathematics like calculus, relations, and functions and other important topics. Hence it becomes important for the readers and learners to revise and view the basic concepts of algebra before moving forward. 2.1. FUNDAMENTALS OF ALGEBRA There is a very significant part of mathematics which is made up by numbers and some general rules of arithmetic and that part is commonly known as ‘algebra.’ Algebra takes the methods and learnings of arithmetic in such a way that it becomes easy to imply the rules related with the calculation of numbers and make use of these rules to work with some symbols too other than the num-bers. The adoption of algebra provides easy access to several other branches of mathematics, rather than an abrupt makeover into new fields, with the use of previously attained knowledge of the use of basic arithmetical operations. The way of writing quantities in some common ways instead of a particular set of arithmetic terms is widely known.
eBook - PDFMaths: A Student's Survival Guide
A Self-Help Workbook for Science and Engineering Students
- Jenny Olive(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
1 Basic Algebra: some reminders of how it works In many areas of science and engineering, information can be made clearer and more helpful if it is thought of in a mathematical way. Because this is so, algebra is extremely important since it gives you a powerful and concise way of handling information to solve problems. This means that you need to be confident and comfortable with the various techniques for handling expressions and equations. The chapter is divided up into the following sections. 1.A Handling unknown quantities (a) Where do you start? Self-test 1, (b) A mind-reading explained, (c) Some basic rules, (d) Working out in the right order, (e) Using negative numbers, (f) Putting into brackets, or factorising 1.B Multiplications and factorising: the next stage (a) Self-test 2, (b) Multiplying out two brackets, (c) More factorisation: putting things back into brackets 1.C Using fractions (a) Equivalent fractions and cancelling down, (b) Tidying up more complicated fractions, (c) Adding fractions in arithmetic and algebra, (d) Repeated factors in adding fractions, (e) Subtracting fractions, (f) Multiplying fractions, (g) Dividing fractions 1.D The three rules for working with powers (a) Handling powers which are whole numbers, (b) Some special cases 1.E The different kinds of numbers (a) The counting numbers and zero, (b) Including negative numbers: the set of integers, (c) Including fractions: the set of rational numbers, (d) Including everything on the number line: the set of real numbers, (e) Complex numbers: a very brief forwards look 1.F Working with different kinds of number: some examples (a) Other number bases: the binary system, (b) Prime numbers and factors, (c) A useful application – simplifying square roots, (d) Simplifying fractions with signs underneath 1.A Handling unknown quantities 1.A. (a) Where do you start? Self-test 1 All the maths in this book which is directly concerned with your courses depends on a foundation of Basic Algebra.
eBook - ePubLearning and Teaching Mathematics
An International Perspective
- Peter Bryant, Terezinha Nunes(Authors)
- 2016(Publication Date)
- Psychology Press(Publisher)
The different interpretations will lead to different ways of tackling the problem and to different solutions: in the former case one finds the roots of the equation using the quadratic formula, in the latter the values of parameters p and q are sought for which the coefficients of the same powers of x in both expressions are equal (p + 2 q = 5 and 3 p - q = 1). From this first section of the chapter, one might be led to conclude that there could be as many different perspectives on algebra as there are researchers one might choose to listen to. However, underlying all these perspectives—even a functional one—one can find a few common threads. All use algebra at least as a notation, a tool whereby we not only represent numbers and quantities with literal symbols but also calculate with these symbols. One could stretch this definition to the extent of saying that algebra is also a tool for calculus (in fact, historically, algebra led to calculus). However, that would miss the point, which is that the symbols have different interpretations depending on the conceptual domain (i.e. the letters represent different objects in school algebra than they represent in the calculus courses). In addition, some new symbols are generated and the procedures for calculating with the symbols also change in subtle ways as one moves from one mathematical sub-discipline to another. Algebra can thus be viewed as the mathematics course in which students are introduced to the principal ways in which letters are used to represent numbers and numerical relationships—in expressions of generality and as unknowns—and to the corresponding activities involved with these uses of letters—on the one hand, justifying, proving, and predicting, and, on the other hand, solving. These uses of algebraic letters all require some translation from one representation to another; furthermore, the activities associated with these letter-uses all require some symbol manipulation
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
F u n d a m e n t a l s o f A l g e b r a S E C T I O N I I I Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 376 OBJECTIVES After studying this unit you should be able to ■ express word statements as algebraic expressions. ■ express diagram values as algebraic expressions. ■ evaluate algebraic expressions by substituting given numbers for letter values. ■ solve formulas by substituting numbers for letters, word statements, and diagram values. Algebra is a branch of mathematics that uses letters to represent numbers; algebra is an extension of arithmetic. The rules and procedures that apply to arithmetic also apply to algebra. By the use of letters, general rules called formulas can be stated mathematically. The expression, 8 C 5 5 s 8 F 2 32 8 d 9 is an example of a formula that is used to express degrees Fahrenheit as degrees Celsius. Many operations in shop, construction, and indus-trial work are expressed as formulas. Business, finance, transportation, agriculture, and health occupations require the employee to understand and apply formulas. A knowledge of algebra fundamentals is necessary in a wide range of occupations. Algebra is often used in solving on-the-job geometry and trigonometry problems. The basic principles of algebra presented in this text are intended to provide a practical back-ground for diverse occupational applications. 12–1 SYMBOLISM Symbols are the language of algebra. Both arithmetic and literal numbers are used in algebra. Arithmetic numbers or constants are numbers that have definite numerical values, such as 2, 8.5, and 1 4 .
eBook - ePub- James J. Kaput, David W. Carraher, Maria L. Blanton, James J. Kaput, David W. Carraher, Maria L. Blanton(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
algebra means the content of traditional algebra I and the courses that follow it. For authors of algebra I textbooks, algebra is a tightly integrated system of symbolic procedures, each of which is closely connected with a particular problem type. The procedures are often introduced as the mathematical means to solve specific types of problems, but the focus quickly becomes learning how to manipulate symbolic expressions. These procedures are then practiced extensively and later applied to specific problem situations (i.e., word problems). Teaching this content involves helping students to interpret various commands—solve, reduce, factor, simplify—as calls to apply memorized procedures that have little meaning beyond the immediate context. For many students, this reduces algebra to a set of rituals involving strings of symbols and rules for rewriting them instead of being a useful or powerful way to reason about situations and questions that matter to them. Consequently, many students limit their engagement with algebra and stop trying to understand its nature and purpose. In many cases, this marks more or less the end of their mathematical growth.Many mathematics educators have recognized the deep problems of content and impact of algebra I and have made introductory algebra a major site in curricular reform efforts (Chazan, 2000; Dossey, 1998; Edwards, 1990; Fey, 1989; Heid, 1995; Phillips & Lappan, 1998). In one class of proposals, algebra is presented as a set of tools for analyzing realistic problems that outstrip students’ arithmetic capabilities. In contrast to algebra I, problem situations involving related quantities serve as the true source and ground for the development of algebraic methods, rather than mere pretext (Chazan, 2000; Lobato, Gamoran, & Magidson, 1993; Phillips & Lappan, 1998). These introductions to algebra aim to develop students’ abilities to use verbal rules, tables of values, graphs, and algebraic expressions to analyze the mathematical functions embedded in the problem situations, and centrally involve computer-based tools and graphing calculators to achieve these goals (Confrey, 1991; Demana & Waits, 1990; Heid, 1995; Schwartz & Yerushalmy, 1992).Other proposals have emphasized the abstract and formal aspects of mathematical practice, suggesting that introductory algebra should develop students’ abilities to identify and analyze abstract mathematical objects and systems. For example, Cuoco (1993, 1995) characterized algebra as the study of numerical and symbolic calculations and, through the development of a theory of calculation, the study of operations, relations among them (e.g., distributivity), and mathematical systems structured by those operations. Cuoco’s proposal reflects mathematicians’ interest in the study of increasing abstract and general algebraic systems.
eBook - ePub- Wayne A. Wickelgren(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
11Problems from Mathematics, Science, and Engineering
This chapter is designed to establish the generality of the problem-solving methods discussed throughout the book. In previous chapters, the problems used to illustrate the methods were deliberately selected so that they could be solved by the reader with no more background than a high school student with one year of algebra and one year of plane geometry. Many of the problems were of the puzzle (or brain teaser or recreational mathematics) variety, which require no specialized knowledge of mathematics, science, or engineering. Although methods for solving such problems have some recreational interest, there is also a serious practical reason in mastering them, for they are also useful for solving serious problems in all areas of mathematics, science, and engineering. This chapter is designed to demonstrate this applicability and to give the reader some experience in it.ALGEBRA
The solution of systems of simultaneous linear equations provides a simple example of the use of evaluation functions, hill climbing, and subgoals. As an example, consider the following system of three linear equations:(E1)(E2)(E3)The operations available for solving such a system are essentially the following. We can (a) multiply both sides of an equation by the same number, (b) add equals to equals (or subtract equals from equals), and (c) substitute equals for equals. As an example of the first, consider the action of multiplying both sides of equation (E2) by the number —2. This yields the equation —2x —4y —10z
eBook - PDFTeaching and Learning Mathematics
A Teacher's Guide to Recent Research and Its Application
- Marilyn Nickson(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 5 The Use of Technology in the Teaching and Learning of Mathematics Introduction Technology appears increasingly in research related to the teaching and learning of mathematics. Studies have been undertaken in connection with various areas of mathematics such as algebra (e.g. Confrey 1994, Cedillo Avalos 1997), geometry (e.g. Mogetta 2000, Olivero 2001) and probability and statistics (e.g. Pratt and Noss 1998). There is also evidence of theorizing in connection with the use of technology in the context of the mathematics classroom (e.g. Nemirovsky et al. 1998, Noss 2002). In this chapter, we shall consider some of these studies together with others in these areas, and will examine research into the effects of the use of technology on pupils in the classroom context. Algebra Algebra is an area of mathematics that attracts a great deal of attention where the use of technology in the teaching and learning of mathematics is concerned, and where it is seen to have great potential (Monaghan 1995). While there are many studies that appear to indicate that there are advantages in using technology in the teaching of algebra, it may be salutary to note Pimm's cautions in this respect. Pimm (1995) argues that Ironically, technology is being used to insist on screen (graphical) inter-pretation of algebraic forms. There is a strong presumption that symbolic forms are to be interpreted graphically, rather than dealt with directly* (p. 104). He points to the promotion of the concept of function in particular through the medium of tech-nology and suggests that the development of this perspective is contrary to the historical development of the subject (see Kieran 1992). He believes that the ultimate effect is to 'switch from description to prescription: from the fact that it is possible to describe one thing in terms of another, to the fact that it must so be seen (Pimm 1995, p. 105).
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
280 Problems, Patterns, and Relations 281 3. Algebra is an art, characterized by order and internal consistency. Children who are involved in algebraic thinking gain a better understanding of the underlying structure and properties of mathematics (Carpenter & Levi, 2000). 4. Algebra is a language that uses carefully defined terms and symbols. These terms and symbols enhance children’s ability to communicate about real-life situations and mathematics itself. 5. Algebra is a tool. In the past, many algebra courses presented algebra as a tool. But too often, students only learned the intricacies of the tool, without learning how to use it in any practical or meaningful way. The NCTM’s Principles and Standards (2000) do not recommend this limited approach to algebra, nor is it the approach we will present here. Algebra is a tool, but one that should be meaningful and useful. Bringing algebra to elementary school does not mean just adding one more topic to a crowded curriculum. It does mean using the mathematics in the curriculum to help chil- dren develop algebraic thinking or reasoning. As Carraher and Schliemann (2010, p. 27) write “The early mathemat- ics curriculum abounds with opportunities for promoting algebraic reasoning.” This chapter begins with a discussion of topics—problems, patterns, and relations—that will be used in the remainder of the chapter to build algebraic thinking. After discussing the algebraic language and symbols appropriate for ele- mentary students, we will look at ways to help children develop algebraic thinking through representing, generalizing, and justifying. Routine problems Routine problems are often considered exercises for practicing computation, but they can also be used to build algebraic understanding. You will find routine problems like these in every textbook: Routine Problem A: Before her birthday, Jane had 8 toy trucks. At her birthday party, she was given some more trucks.
- Helen Forgasz, Gabriele Kaiser, Mellony Graven(Authors)
- 2018(Publication Date)
- Springer Open(Publisher)
This is true for educations and professions as engineering, economy, computer science, and natural sciences as physics, chem-istry or biology. The responsibility for teaching students the algebra they need for further education and professions lies with the school. If this is not provided in school, it will in fl uence student ’ s possibilities to pursue a number of educations based on their home background (Gr ø nmo 2015 ). This is not in accordance with the goal of equity for access to educations and later professional work that are a main goal in education in so many countries (Ibid.). The school ’ s responsibility for providing this type of knowledge to their stu-dents ’ is therefore closely related to students ’ equal rights to education in a changing world, and we have to take into account the direction of development in the society (OECD 2017 ). There is an ongoing discussion about the need for people with creativity and competence in how to handle changes in many countries including Norway. On the other hand, there seems to be less discussion about the need to emphasise students ’ learning of basic knowledge in the mathematical language algebra, needed in so many professions. I will argue that basic knowledge in algebra is more to be seen as complementary and necessary for being creative, rather than something opposing creativity. This may not be true for all types of societies, but at least for the highly developed technological society we have in many countries today. Without the language to develop technology and science, creativity is probably not very helpful. Algebra was probably not that important for so many fi fty years ago as it is today. But taken into account the changes and challenges we are facing in a modern society (Ibid.), competence in algebra is essential and for that reason also an issue of importance from the perspectives of giving all students the possibility to pursue the education and job positions they want.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
