Rational Numbers and Fractions

7 Key excerpts on "Rational Numbers and Fractions"

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  eBook - ePub

  Teaching Fractions and Ratios for Understanding

  Essential Content Knowledge and Instructional Strategies for Teachers

  • Susan J. Lamon(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  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. Other important distinctions related to the rational numbers are made in the following examples.

  • All rational numbers may be written in fraction form.

  3 4

  ,    

  4 3

     

  (

  usually   written   as    

  2 3

  )

  ,    

  2.1 4.1

     

  (

  usually   written   as    

  21 41

  )

  ,   and    

  1 2

  1 4

  (

  usually   written   as      

  2 1

  )

  are all fractions and rational numbers.

  • Not all numbers written in fraction form are rational.

  π 2

  is not a rational number although it is written in fraction form.

  • Each fraction does not correspond to a different rational number.

  There is not a different rational number for each of the three fractions

  2 3

  ,    

  6 9

  ,     and    

  10 15

  . Just as one and the same woman might be addressed as Mrs. Jones, Mom, Mother, Maggie, Dear, Aunty Meg, and Margaret, these fractions are different numerals designating the same rational number. A single rational number underlies all of the equivalent forms of a fraction.

  • Rational numbers may be written as fractions, but they may be written in other forms as well.

  Terminating decimals are rational numbers. Non-terminating, repeating decimals are rational numbers. Percentages are rational numbers. Non-terminating, non-repeating decimals are not rational numbers. Ratios and rates are rational numbers.

  Fractions as Numbers

  When we speak of a fraction as a number, we are really referring to the underlying rational number. Understanding a fraction as a number entails realizing, for example, that

  1 4

  refers to the same relative amount in each of the following pictures. There is but one rational number underlying all of these relative amounts. Whether we call it

  1 4

  ,

  4 16

  ,

  3 8

  , or    

  2 8

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  Effective Teaching Strategies for Dyscalculia and Learning Difficulties in Mathematics

  Perspectives from Cognitive Neuroscience

  • Marie-Pascale Noël, Giannis Karagiannakis(Authors)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  6 Rational numbers

  DOI: 10.4324/b22795-6

  A rational number is the quotient of two integers. Rational numbers are presented in three different notations: fractions (1/4), decimal numbers (0.25), and percentages (25%). During the last years of the primary school cycle (around fourth or fifth grade), the child is introduced to rational numbers. Mastering these rational numbers is important for school success already in primary schools, but also beyond. Thus, the knowledge of fractions is essential for more advanced mathematical achievement such as algebra, but is also a predictor of the ability to participate in the labour force in the United States (Geary, Hoard, Nugent & Bailey, 2012; Siegler & Pyke, 2013). Indeed, it has been observed that poor mastery of rational numbers prevents access to many middle- and high-income jobs (McCloskey, 2007). However, this learning is a problem for many students, including those in middle school or high school, mainly because they do not understand the meaning of these rational numbers.

  6.1 Learning rational numbers: a challenge!

  Learning rational numbers is a real challenge for children. According to a study carried out by the French Community of Belgium (1996, unpublished), major difficulties persist in this learning among students in seventh or eighth grades (around the age of 13). For example, about 1 out of 4 students is unable to find a number between 2 decimal numbers (e.g., between 72.4 and 72.5), and half of them are unable to calculate a percentage (e.g., 21% of 6,200). Similarly, among a large sample of college students in the United States, only 33% correctly identified the largest of four fractions (Stigler, Givin & Thompson, 2010). One of the main reasons for this challenge is that children are facing a real conceptual shift, and the mistakes they make reveal misconceptions about rational numbers. Indeed, in many cases, the errors can be explained by the fact that students tend to view fractions and decimals as whole numbers and to apply to them the rules previously used on natural numbers. The interference of prior knowledge when learning rational numbers seems to be one of the main sources of difficulties and has been identified in the literature as the natural number bias

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  Fostering Children's Mathematical Power

  An Investigative Approach To K-8 Mathematics Instruction

  • Arthur Baroody, Arthur J. Baroody, Ronald T. Coslick(Authors)
  • 1998(Publication Date)
  • Routledge

    (Publisher)

  We represent such division situations with a remainder sign (e.g., 10 + 3 = 3 r1). In other situations, such as in Problem D in Part III of Investigation 9.1, though, a remaining item can be subdivided. To represent division situations in which the answer is not a whole number and there is no remainder, we need the rational numbers. 1 For example, the solution to Problem D can be represented by. Rational-Number Concepts and Symbols Rational-Number Concepts or Meanings. What meanings can rational numbers have? Four primary meanings have been identified: part of a whole, ratio, quotient, and operator (Kieren, 1988). 2 • Part of a Whole. The term fraction comes from the Latin fractio (a derivative of frangere meaning to break) and, thus, can refer to something broken into parts. More specifically, common fractions can represent a part-whole relationship: a number of parts (represented by the top term or the numerator) of a whole divided into equal-sized parts (the bottom term or the denominator) 3 What is involved in learning this meaning of fractions is developed further in this chapter. • Quotient. A fraction can also represent division or a quotient meaning. For instance,, can be used to represent three dollars divided fairly among four people. A quotient meaning of fractions is explored further in this chapter. • Ratio. A fraction such as can also represent a ratio. A ratio can be represented in other ways such as 4:5. It describes a relationship between two quantities, which may or may not be a part-whole relationship. For example, the ratio 4:5 can represent either (a) four of the five marbles were blue (a part-whole relationship) or (b) there were four blue marbles for every five white marbles (not a part-whole relationship). Chapter 12 discusses various ways ratios can be used to compare quantities. • Operator. An operator meaning can be thought of as a function that involves multiplying by a fraction

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  Teaching Mathematics in Primary Schools

  Principles for effective practice

  • Robyn Jorgensen(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  CHAPTER 11RATIONAL NUMBER

  Topics that fall under the heading of rational number include fractions, decimals, ratio and proportion, rate and per cent. These topics are linked mathematically, but conceptually they are subtly different. In this chapter, key ideas associated with the topics of fractions, decimals, ratio, rate, proportion and per cent and their interlinked nature are presented. Approaches for enhancing students' knowledge of these topics are described.

  Common and decimal fractions

  Fractional numbers can be represented in fraction form (e.g.¼) and in decimal form (e.g. 0.25), and the terms 'common fraction' and 'decimal fraction', respectively, are used to distinguish the two symbolic representations. The word fraction is frequently applied to numbers in both fraction form and decimal form, yet there are subtle conceptual differences between common fractions and decimal fractions. Common fraction understanding is based on the part-whole concept. Decimal fraction understanding stems from a combination of an understanding of common fractions, and whole number and place value knowledge. For simplification, in this chapter common fractions are referred to as 'fractions', and decimal fractions as 'decimals'.

  Whole number and rational number connections

  Whole number understanding provides the foundation for understanding of rational numbers. Particular rational number topics provide a foundation as well as a link to other rational number topics. Decimal understanding is connected to both fraction and whole number knowledge. Ratio and proportion understanding links to fractions, as well as to multiplicative thinking developed through the study of whole numbers. Rate links to ratio. Per cent links to decimals and fractions, and to ratio and proportion. The interconnected nature of rational number topics to each other and to whole number is depicted in the accompanying flowchart. New Zealand's Number Framework (Ministry of Education, New Zealand 2008) includes nine global stages of number knowledge and strategy understanding that encompass the development of rational number knowledge.

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  eBook - ePub

  Numbers

  Histories, Mysteries, Theories

  • Albrecht Beutelspacher, Andrea Bruder, Andrea Easterday(Authors)
  • 2016(Publication Date)
  • Dover Publications

    (Publisher)

  For example, to determine the greatest common divisor of 101 and 35, we may proceed as follows: Let us imagine a line segment of length 101 and remove from it a line segment of length 35 as many times as possible (in this case, that is twice). The remainder is a line segment of length 31.

  In a second step, we take the line segment of length 35 and remove a line segment of length 31; we are left with a remainder of 4. Now remove a line segment of length 4 from the line segment of length 31 seven times. The remainder is of length 3. Upon removing a line segment of length 3 from the line segment of length 4, we obtain a remainder of length 1. Since the latter divides the line segment of length 3, we have found a common measure of 1 for all the line segments we considered, and in particular for the line segments of length 101 and 35.

  Today we write these geometric considerations as follows:

  Using the Euclidean algorithm, we may easily determine the greatest common divisor of two very large numbers, without knowing their prime factorization. This technique plays a key role in many applications, including modern cryptography (see Chapter 1 ).

  3.3 Rational numbers

  Natura non facit saltus.

  One can introduce fractions in several different ways: as portions, as fragments, or as proportions. In contemporary school books, one sees an assortment of models for fractions: pieces of cake or pizza, parts of rectangles and line segments.

  We’ve become accustomed to referring to fractions as “rational numbers.” This term comes from the Latin word ratio , which among other things means proportion. In the literal sense “rational numbers” are ratios, but they are the same as fractions.

  Whole numbers are also fractions, because the number 5 for example can be written as . This means the set of whole numbers is included in the set of fractions.

  Calculations work wonderfully with rational numbers; for computation they leave hardly a wish unfulfilled. One can add them, subtract them, multiply and divide. These operations work as well as one might ask.

  Adding and subtracting rational numbers is particularly easy, at least if we imagine a rational number as a distance, a “piece of cake,” or any other suitable quantity. In order to determine , we consider a distance of length and a distance of length and join the two pieces together. The result is a distance of length

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  Axiomatic Set Theory

  • Patrick Suppes(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 6

  RATIONAL NUMBERS AND REAL NUMBERS*

  § 6.1 Introduction. To show that our axioms for set theory are adequate to permit the systematic development of classical mathematics, it is not sufficient merely to construct the natural numbers as we did in the previous chapter. At the very least we need to show that we can construct entities which have all the expected properties of the real numbers.

  The two basic set-theoretical methods of constructing the real numbers out of the natural numbers are due to Cantor and Dedekind, but Bertrand Russell also deserves credit for making clear the exact character of these constructions and for being completely explicit about identifying the real numbers as the constructed entities.

  Antecedent to the construction of the real numbers is the construction of the rational numbers (intuitively a rational member is a number which can be represented as the ratio of two integers). Several alternative courses of development can be followed:

  Several variants of these three courses are possible depending upon the choice of a level at which to introduce negative numbers. Course III will be adopted here. Non-negative fractions are defined as ordered pairs of non-negative integers. Thus the fraction = 〈1,2〉. Then non-negative rational numbers are defined as certain equivalence classes of fractions. For instance, the non-negative rational number corresponding to the fraction is the set of all fractions m/n such that n = 2m. To get to all rational numbers, we go up another level of abstraction. We say that two ordered pairs 〈x, y〉 and 〈u, v〉 of non-negative rational numbers are equivalent when

  x + v = y+ u,

  and a rational number is just an equivalence class of such ordered pairs. Perhaps it may seem odd to distinguish between the fraction the non-negative rational number , the ordered pair 〈 , 〉, and the rational number [〈 ,

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  Rational Numbers

  An Integration of Research

  • Thomas P. Carpenter, Elizabeth Fennema, Thomas A. Romberg(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  What orientations should a researcher or teacher take away from the analyses in the previous sections ? First, it has been argued that as a result of seeing mathematics as a human activity, one can conclude that rational number thinking, knowing, and understanding, although related to whole number knowing, are distinct from and not dependent on it. It is better to see such forms of number knowing as intertwined.

  In particular, there are two critical complementarities that are seen in the mathematics and in the domains of application of rational numbers. The first is that rational numbers are at once quotients and ratios — extensive and intensive quantities. The second is that in the applicational actions of a knower, there are the complementary dimensions of the subconstructs — at least quotient, measure, operator, and ratio numbers.

  The first of the complementarities described earlier is related logically to the existence of the fractions for any nonzero integer n. Thus, the unit fractions along with the quotient and ratio nature form a mathematical base for rational numbers. This base also is seen in the mathematical actions of young persons. Considering rational numbers as a humanly knowable activity means taking into account the properties indicated and the many others that distinguish rational from natural numbers.

  Seeing humans as capable of knowing rational numbers brings into play ideas from two models. The first argues for the efficacy of seeing rational number knowing as one of four types: ethnomathematical, intuitive, technical symbolic, and axiomatic deductive. The challenge is to find the interrelationships among these knowing types and their appropriate places in the curriculum. The second model, a more general model of the growth of mathematical understanding, portrays an interweaving, nonlinear leveled structure. Each level is characterized by a complementarity of process and form. Because of this, outer-level knowing can derive from either the processes or the forms of inner levels. This model also suggests four bases for mathematical knowing: action, image, form, and structure. These are not to be confused with traditional concrete, pictoral, and symbolic modes. Under this model, understanding rational numbers is characterized as a dynamic whole, of knowing rational numbers at many levels at once.

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