Addition and Subtraction of Rational Expressions
The addition and subtraction of rational expressions involves combining or separating algebraic fractions. To add or subtract rational expressions, you need to find a common denominator, then perform the operation on the numerators while keeping the common denominator. Simplifying the resulting expression is often necessary to obtain the final answer.
6 Key excerpts on "Addition and Subtraction of Rational Expressions"
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- 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 - ePubEffective Teaching Strategies for Dyscalculia and Learning Difficulties in Mathematics
Perspectives from Cognitive Neuroscience
- Marie-Pascale Noël, Giannis Karagiannakis(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...When the denominators are different, a common denominator must first be found and one or both fractions must be transformed into equivalent fractions with the same denominator. This notion of equivalence of fractions is a real challenge for many students. In their longitudinal study of students in third to sixth grades, Jordan et al. (2017) observed that good multiplication skills with natural numbers make it easier for children to recognize equivalent fractions (e.g., 1/2, 2/4, 8/16) and that good division skills with whole numbers make it easier to simplify fractions (e.g., simplifying a fraction such as 14/35 into 2/5). If in the case of the addition or subtraction of fractions it is necessary to transform the terms of the problem into equivalent fractions with a common denominator and then add or subtract their numerators and retain the common denominator in the answer, the same is not true for multiplication. Indeed, it is not necessary to find a common denominator and transform the terms since the numerators and denominators will simply be multiplied (3/5 × 4/5 = 12/25). However, some children produce errors such as ‘3/5 × 4/5 = 12/5’ (Siegler et al., 2011), interpreted by Siegler and Pyke (2013) as operation errors in which a procedural step in one arithmetic operation is mistakenly used for another operation. In the case of fraction multiplication, these 2 authors also observed such operation errors in sixth and eighth graders (11 and 13 years old) on 46% of the problems. To divide one fraction by another, invert one of the fractions, then multiply the numerators together and the denominators together. This procedure remains very opaque for many students, probably in part, because it is also misunderstood by most teachers themselves (Ball, 1990). Siegler and Pyke (2013) found that sixth and eighth graders made operating errors on 55% of fraction division problems...
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...However, if we are teaching so that operations arise naturally from a very deep understanding and fraction sense, we need to be aware of a broad range of phenomena that are the sources of meaning underlying those fraction symbols. Understanding rational numbers involves the coordination of many different but interconnected ideas and interpretations. Unfortunately, fraction instruction has traditionally focused on only one interpretation of rational numbers, that of part–whole comparisons, after which the algorithms for symbolic operation are introduced. This means that student understanding of a very complex structure (the rational number system) is teetering on a small, shaky foundation. Instruction has not provided sufficient access to other ways of interpreting a b : as a measure, as an operator, as a quotient, and as a ratio or a rate. We will address all of these interpretations later. Having a mature understanding of rational numbers entails much more than being able to manipulate symbols. It means being able to make connections to many different situations modeled by those symbols. Part–whole comparisons are not mathematically or psychologically independent of other meanings, but to ignore those other ideas in instruction leaves a child with a deficient understanding of the part–whole fractions themselves, and an impoverished foundation for the rational number system, the real numbers, the complex numbers, and all of the higher mathematical and scientific ideas that rely on these number systems. I reject the use of the word fraction as referring exclusively to one of the interpretations of the rational numbers, namely, part–whole comparisons. Because the part–whole comparison was usually the only meaning ever used in instruction, it is understandable that fraction and part–whole fraction became synonymous...
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...
eBook - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...It is important to provide opportunities for students to engage with the big ideas of rational number topics so that they can both see connections between these topics and understand key ideas within each topic. Concrete materials need to be selected carefully on the basis of their value in accurately representing key concepts for rational number understanding, and the associated language must become a natural part of students' dialogue. Confusion with rational number topics often occurs as a result of rushing to symbolic representation and manipulation of numbers before a rich conceptual framework has been developed sufficiently. Rational number study needs to be an integral part of early mathematics classrooms in an exploratory sense, so that a solid foundation is laid for more formal investigations in the middle and later years.Review questions11.1Describe rational numbers in terms of their location on a number line. Outline how rational numbers extend students' conceptualisation of numbers on a number line.11.2Why is decimal knowledge only partly dependent on fraction knowledge?11.3Describe how a fraction meaning may interfere with developing ratio meaning.11.4Outline the multiplicative structure of ratio and percentage increase situations.Further readingCarpenter, T.P., Fennema, E. & Romberg, T.A. (2012).Rational numbers: An integration of research.New York, NY: Routledge.Condon, C. & Hilton, S. (1999). Decimal dilemmas.Australian Primary Mathematics Classroom,4(3), 26-31.Cramer, K. & Bezuk, N. (1991). Multiplication of fractions: Teaching lor understanding.Arithmetic Teacher,39(3), 34-7.Eames, C. & Barker, M. (2011). Understanding student learning in environmental education in Aotearoa New Zealand.Australian Journal oj Environmental Education.27(1), 186-91.Empson, S. (2001). Equal sharing and the roots of fraction equivalence.Teaching Children Mathematics,7(7), 421-5.Graeber, A.O. & Baker, K.M. (1992)...
