Algebraic Fractions
Algebraic fractions are expressions that involve both algebraic terms and fractions. They are formed by dividing one algebraic expression by another and can include variables in the numerator and/or denominator. Simplifying algebraic fractions involves factoring, canceling common factors, and applying the rules of operations for fractions. These fractions are commonly used in solving equations and simplifying complex expressions.
7 Key excerpts on "Algebraic Fractions"
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...The second is to show that algebra greatly aids us in describing the relations between numbers and, by extension, physical quantities. The third is to show you how to tackle algebraic problems. 2.1 Arithmetic with fractions Any number we can write down, punch into a calculator, or type into a spreadsheet on a computer is a rational number; that is, it can be written as a fraction. Any integer is a fraction, n = n /1, and furthermore any decimal number that can be displayed is a fraction, for example a number like 3.14 can be regarded as a shorthand for 314/100. Arithmetic has been completely mastered when we can perform all the basic operations with fractions. And once we can do arithmetic with fractions, we can handle Algebraic Fractions. Practice with both the arithmetic and the algebra of fractions is provided in the End of Chapter Questions. The fundamental fraction is the reciprocal of an integer The key to understanding fractions is first to understand what is meant by the reciprocal of an integer. The reciprocal, 1/ n, of any integer, n, is defined by the fact that if we add up n of them, or equivalently if we multiply 1/ n by n, we are back to 1 whole. It is the mathematical equivalent of pie slices (hence pie charts as in Chapter 9). If we slice a pie into 21 pieces, each is 1/21st of a pie. Once we have the reciprocal, fractions follow immediately, they are just integers multiplied by reciprocals; that is, if a and b are integers the general fraction can be written as a b ≡ a × 1 b ≡ 1 b × a. (EQ2.1) The reciprocal of a product is the product of the reciprocals To take an example suppose we want 21 slices. We can get to our 21 slices in at least three ways. We could just set about slicing 21 pieces. We could first cut the pie into 7 equal slices and then divide each of these into 3. Finally, we could first cut 3 slices and then divide each of these into 7...
eBook - ePub- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...Fractions When you perform algebraic manipulations, you soon encounter fractions, which means parts of numbers. We all learned to manipulate fractions in school, but in these days of computers and calculators, many people have forgotten how to do this. Remembering how to multiply and divide fractions causes particular problems. All fractions have three components – a numerator, a denominator and a division symbol: The division symbol in a simple fraction indicates that the entire expression above the division symbol is the numerator and must be treated as if it were one number, and the entire expression below the division symbol is the denominator and must be treated as if it were one number. The same order of operations (BEDMAS) applies to fractions as to other mathematical terms. Brackets instruct you to simplify the expression within the bracket before doing anything else. The division symbol in a fraction has the same role as a bracket. It instructs you to treat the quantity above (the numerator) as if it were enclosed in a bracket, and to treat the quantity below (the denominator) as if it were enclosed in another bracket: In a simple fraction, the numerator and the denominator are both integers (whole numbers), e.g. A complex fraction is a fraction where the numerator, denominator or both contain a fraction, e.g. To manipulate (e.g. add, subtract, divide or multiply) complex fractions, you must first convert them to simple fractions. A compound fraction, also called a mixed number, contains integers and fractions, e.g. As with complex fractions, to manipulate compound fractions, you must first convert them to simple fractions. No fraction (simple, complex or compound) can have a denominator with an overall value of zero. This is because, if the denominator of a fraction is zero, the overall value of the fraction is not defined, since you cannot divide by zero...
eBook - ePub- Bernice Kastner(Author)
- 2019(Publication Date)
- Austin Macauley Publishers(Publisher)
...Fractions to be added must refer to the same subdivision of the underlying unit because we can only add the same kinds of things. This is the reason we have to express the fractions in terms of a common denominator, in spite of the complicated and inconvenient nature of the process. Because of the temptation to add the numerators and add the denominators, it would be better not to introduce the set model until after children have had considerable experience in addition and subtraction of fractions that originate in measurement contexts. Algebraic Fractions When so many school children seem to have no understanding of meaning for numerical fractions, as documented by the Siegler and Lortie-Forgues article cited earlier in this chapter, it is hardly surprising that such students find Algebraic Fractions even more troublesome. It is once again the focus on counting and lack of attention to the “continuous” aspect of measurement settings that make it hard for children to develop useful intuitions of fractions. There are also language issues. Recalling the many different meanings for the word “term” and the use of letters to represent either variables or constants shown in chapter 1, we can see that early work in algebra must fill in the gaps created for students who have simply learned arithmetic as a set of “how to do it” procedures. Algebra requires the recognition of generalizations based on meaningful connections and structures, and many children who are competent with procedures have not been prepared to make the transition to the abstractions needed in order to make sense of algebra. The fraction bar and the slash (/) One issue in the context of fractions that are expressed in terms of symbols as “placeholders” for numbers emerged after the invention of typewriters, when people found it convenient to use a “slash” in place of a fraction bar in order to fit the fraction on a single line of typescript...
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...As we know, learning in contexts is a complex and lengthy process, so “saving” children from decimals and percentages until much later is a poor excuse that only cuts short their experiences. Most compelling is the fact that research has shown that by third grade, some children have already developed preferences for expressing quantities as decimals or percentages. (In this book you will see some of their work.) What Are Fractions? Today, the word “fraction” is used in two different ways. First, it is a numeral. Second, in a more abstract sense, it is a number. First, fractions are bipartite symbols, a certain form for writing numbers: a b. This sense of the word fraction refers to a form for writing numbers, a notational system, a symbol, a numeral, two integers written with a bar between them. Second, fractions are non-negative rational numbers. Traditionally, because students begin to study fractions long before they are introduced to the integers, a and b are restricted to the set of whole numbers. This is only a subset of the rational numbers. The top number of a fraction is called the numerator and the bottom number is called the denominator. The order of the numbers is important. Thus, fractions are ordered pairs of numbers, so 3 4 is not the same as the fraction 4 3. Zero may appear in the numerator, but not in the denominator. All of these are fractions in the sense that they are written in the form a b : − 3 4, π 2, 4 2, − 12.2 14.4, 1 2 1 4 However, they are not all fractions in the second sense of the word. Therefore, I will say fraction form when I mean the notation, and fraction when I mean non-negative rational numbers. Rational Numbers Although many people mistakenly use the terms fractions and rational numbers synonymously, they are very different number sets...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...2 FRACTIONS WHAT YOU WILL LEARN • The equivalence of fractions in special cases, and how to compare fractions by reasoning about their size • How to express a fraction as an equivalent fraction with a different denominator; • How to decompose fractions to justify their sum or difference • How to add and subtract fractions with unlike denominators (including mixed numbers) • How to interpret a fraction as a division of the numerator by the denominator • Solve word problems involving division of whole numbers, leading to answers in the form of fractions or mixed numbers • How to apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction • How to interpret and compute quotients of fractions, and solve word problems involving division of fractions SECTIONS IN THIS CHAPTER • Equivalent Fractions • Adding Fractions • Subtracting Fractions • Multiplying Fractions • Dividing Fractions • Word Problems DEFINITIONS Fraction A number that represents part of a whole, part of a set, or a quotient in the form, which can be read as a divided. by b. Denominator The quantity below the line in a fraction...
eBook - ePubRtI in Math
Evidence-Based Interventions
- Linda Forbringer, Wendy Weber(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...While concrete and pictorial representation is used in Tier 1, core materials often move too quickly to abstract representation before learners who struggle with mathematics are able to fully grasp the concepts (Gersten et al., 2009 ; van Garderen, Scheuermann, Poch, & Murray, 2018). In this chapter, we will discuss ways to incorporate these intensive intervention strategies when introducing rational numbers. Fractions Developing Fraction Concepts Fractions present one of the greatest challenges students encounter. National and international test results reveal that American students have consistently struggled with basic fraction concepts (NMAP, 2008, 2019; Siegler, 2017). Understanding fraction concepts is necessary to perform meaningful computations with fractions, and fractions are a pre-requisite for decimals, percent, ratio and proportion, and algebra. Knowledge of fractions in fifth grade predicts student's math achievement in high school, even after controlling for the student's IQ, knowledge of whole numbers, and family education level or income (Siegler, 2017). Even students who have not experienced previous mathematical difficulty can be challenged by fractions. For students with a history of mathematical difficulty, the problem is magnified. To understand fractions, students must master a few big ideas. First, fractional parts are formed when a whole or unit is divided into equal parts. In other words, to understand a fraction, students first need to identify the unit and then make sure it is divided into equal parts. Students who struggle with fractions sometimes miss the importance of having equal parts. The concept of unit can also confuse students, because the word has several different mathematical applications. The smallest piece in base-ten blocks is sometimes called a unit block. For fractions, the unit is the whole object, set, or length that is divided into equal parts...
eBook - ePubThe Problem with Math Is English
A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics
- Concepcion Molina(Author)
- 2012(Publication Date)
- Jossey-Bass(Publisher)
...Chapter Eight Operations with Fractions The misconceptions and assumptions that permeate fractions only multiply when computation comes into play. As in other areas, methods for teaching the operations of fractions typically rely on shallow rules and procedures. The old adage “Ours is not to reason why; just invert and multiply” is yet another example of students learning a process without any idea of the conceptual basis for it. At the same time, the procedures for computing fractions and the results they produce can seem foreign to students used to working with whole numbers. Mystified and lost, students can feel like strangers in a strange land. To help students adjust to operations with fractions, teachers need to provide both a conceptual foundation that connects to whole number operations and guidance in interpreting the language and symbolism. Adding and Subtracting Fractions Most teachers and students would probably consider addition and subtraction to be the easiest of the four basic operations. It is somewhat of a paradox then that many students find adding and subtracting fractions to be extremely challenging. Sometimes, though, what we know gets in the way of learning something new. The habits instilled in students when they add and subtract whole numbers, coupled with an inattention to the language and symbolism of math in instruction, can interfere with students' ability to learn the same two operations with fractions. So What's the Problem? Examine Box 8.1, which illustrates a common error students make when adding fractions with unlike denominators. No doubt, countless teachers have been frustrated by trying to help students avoid this mistake. Box 8.1: Incorrect Addition of Fractions The primary culprit behind this error is the lack of instructional emphasis on the property that only like items can be combined. Students are taught they can only add and subtract fractions with common denominators, but not why...
