Addition and Subtraction of Rational Expressions

3 Key excerpts on "Addition and Subtraction of Rational Expressions"

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  eBook - ePub

  Mathematics Content for Elementary Teachers

  • Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
  • 2004(Publication Date)
  • Routledge

    (Publisher)

  Certainly the rule about multiplying denominators could be used when unlike denominators are not relatively prime, but the product would not be the least common denominator. The solution would be a fraction with a numerator and denominator that have a common factor. If you insist that all common factors be divided out, a simplification step would have to be performed. The following examples should help:

  or Your Turn   7.  Do each of these problems, showing basic intermediate steps:

      

      

  Mixed Numbers

  One type of subtraction exercise involving fractions deserves special consideration because of the regrouping that must be done. Consider . Writing this exercise vertically introduces an alignment that may appear strange . You need to express 4 in a different manner so that the problem will look like one you know how to handle. Although 4 can be written in a multitude of ways, we are going to write it as 3 + 1 and express the 1 as .

  We want to use as an equivalent fraction for 1 because it ensures that the fractions have the same denominator. This changes the problem to a familiar form, .

  The problem could be solved in a horizontal manner, as follows. How to complete an exercise is often a matter of personal preference. We hope that you will practice several different methods, rather than rely on the method you find most comfortable:

  There is another way to look at this problem. Your experience with missing addends should help you reason that is less than one , or that must be added to to generate . This line of reasoning is extended when the sum (4 in this case) is made smaller, meaning that the missing addend would include the missing and the remaining 3 from the sum, or . Although it is unlikely that someone would use this process when working with pencil and paper, it is a common technique for mental arithmetic.

  A final situation involves an exercise in which both the sum and the given addend are mixed numbers with like denominators. In this special case, the fraction part of the sum is less than the fraction part of the given addend. You will need to do some regrouping before the subtraction can be completed. If the exercise is , the first thing to do is rename the sum so that you can regroup, , which is . Now the exercise can be expressed as

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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)

  In the investigative approach, a teacher might pose problems and encourage children to invent their own strategies for operating on fractions. This could serve as the conceptual basis for them later reinventing formal algorithms. To see how and to perhaps deepen your own understanding of operations on fractions, complete the three parts below either on your own or, preferably, with the help of your group. Discuss your solutions with your group or class.

  Part I: Addition and Subtraction

  1. Can you add apples and oranges? How is adding (or subtracting) unlike fractions comparable to adding apples and oranges?
  2. Which of the following is assumed to be true when fractions are added or subtracted (e.g., or )? Circle the letter of any correct statement.
     1. Each addend can represent a fraction of different-sized wholes.
     2. Each addend can represent a fraction of same-sized wholes.
     3. Each addend can represent a fraction of the same whole.
     4. None of the above.
     5. Statements a, b, and c are all true.
     6. Both statements b and c are true.
  3. Put yourself in the place of an elementary-level child who does not know the algorithm for adding unlike fractions. Illustrate how you could use your knowledge of whole-number addition and manipulatives such as Cuisenaire rods, Fraction Circles, or Fraction Tiles to model and to solve Problem A below.
     Problem A: A Treat Tragedy (◆ 3–5). Mary made a special treat for her birthday party—a tuna dip in the shape of a bunny. Ruffus her dog discovered the treat and ate one-half of it. Later he polished off another one-third of the (original) treat. What fraction of the bunny-shaped treat did Ruffus wolf down altogether?
  4. A common solution children devise for Problem A is shown below. How could a teacher create cognitive conflict in order to prompt children to reconsider this incorrect solution and to discover a correct method for adding and with the rods?

  Part II: Groups-of and Area Meanings of Multiplication

  Put yourself in the place of a child who does not know a fraction-multiplication algorithm. Illustrate how you could informally model and solve Problems B, C, D, and E using manipulatives such as Cuisenaire rods

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  The 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. By connecting to the role of like items in combining fractions, teachers can help students build on the conceptual foundation of whole-number addition and subtraction. Chapter Five, in the discussion on the order of operations, explains how math instruction initially emphasizes the property of like items in basic addition and subtraction. Students receive problems such as 2 apples + 3 apples = 5 apples, while also being told they can't add 2 apples to 3 oranges without converting to a common unit, such as fruit

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