Fractions in Expressions and Equations
Fractions in expressions and equations involve using numerical values in the form of fractions to represent parts of a whole within mathematical operations. This includes adding, subtracting, multiplying, and dividing fractions within algebraic expressions and equations. Understanding how to work with fractions in these contexts is essential for solving a wide range of mathematical problems.
8 Key excerpts on "Fractions in Expressions and Equations"
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)
...With that in mind, Box 7.2 contains a generic definition of the concept of fraction. Box 7.2 Fraction (n., mathematics). A generic expression in mathematics with multiple interpretations that is typically represented as, where a and b are natural numbers. The possible meanings are dependent on the context and include an expression of a relationship, an expression of a quantity, the operation of division, and a subset of the real numbers. Because of the tendency to gravitate to the “part of a whole” notion of a fraction, there is a focus on the universal nature of the term in order to clearly grasp the “big picture” of fractions. Although the U.S. mathematics education system does gradually cover other uses of fractions than parts of a whole, students rarely receive the big picture to serve as a foundation. As a result, when confronted with other uses, such as ratio and probability, students often struggle to wrap their minds around them. A solid definition and visual overview of fractions provide this needed foundation and make up for its lack in mathematics textbooks, standards, and traditional instruction. A conceptual understanding of fractions requires more than a clear grasp of the big picture. Essential to a deep understanding of fractions is thorough knowledge of the constituent parts. Thus, the use of precise and consistent definitions for the terms associated with the components of the fraction kingdom, such as ratio and slope, are necessary so that students do not mistakenly assign the characteristics of one term to another. In science, students are expected not to make this type of mistake, such as confusing the characteristics of birds with those of insects. Mathematics should be no different. An examination of each part of the fraction kingdom is in order. Fractions as Relationships The first category in the fraction kingdom is the use of fractions to express relationships...
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...
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...
- 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 - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...The whole in this sense is more than one, but the collective set of objects must be regarded as one whole in order to operate with the objects in a fraction sense. Student representing one quarter One of the other crucial understandings of fractions relates to fractions as division situations, and this is essential for working with algebraic calculations. Other big fraction ideas are that fractions are ratios, and that fractions are operators (fractions as ratio are addressed under 'Ratio and proportion' later in this, chapter). Fractions as operators refers to the way the numerator (the top number) in a fraction is that which multiplies, and the denominator (the bottom number) is that which divides. Knowing fractions as operators should be a natural by-product of a rich conceptual understanding of fractions; however, it is not elaborated on here. In this section, ideas for developing students' understanding of the following three notions are the focus: fractions as part of a whole fractions as part of a set fractions as division. Intuitive fraction ideas Many students come to school with intuitive traction ideas. The most commonly used fraction is one-half, and children are exposed to this from an early age through natural language and experience. When cutting toast or sandwiches, children are often asked if they want their bread cut in half or if they want it left whole. They may be cajoled into eating half their vegetables in order to get ice-cream. When they are given a bag of sweets, they are told to give half to their sibling. The notion of one-half is usually well established, but mathematically, it may not be as precise as required. A young student, provided with a diagram of a rectangle cut into three equal parts, two of which were shaded, was asked to state how much of the rectangle was shaded. The response was 'one-half'...
eBook - ePubKnowing and Teaching Elementary Mathematics
Teachers' Understanding of Fundamental Mathematics in China and the United States
- Liping Ma(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Most of the teachers generated at least one correct and appropriate representation. Their ability to generate representations that used a rich variety of subjects and different models of division by fractions seemed to be based on their solid knowledge of the topic. On the other hand, the U.S. teachers, who were unable to represent the operation, did not correctly explain its meaning. This suggests that in order to have a pedagogically powerful representation for a topic, a teacher should first have a comprehensive understanding of it. Notes 1 As indicated earlier, 21 of the 23 teachers attempted the calculation. 2 According to the current national mathematics curriculum of China, the concept of fractions is not taught until Grade 4. Division by fractions is taught in Grade 6, the last year of elementary education. 3 In China, the rule of “maintaining the value of a quotient” is introduced as a part of whole number division. The rule is: While the dividend and the divisor are multiplied, or divided, by the same number, the quotient remains unchanged. For example, 15 ÷ 5 = 3, so (15 × 2) ÷ (5 × 2) = 3 and (15 ÷ 2) ÷ (5 ÷ 2) = 3. 4 In the Chinese national curriculum, topics related to fractions are taught in this order: Introduction of “primary knowledge about fractions” (the concept of fraction) without operations. Introduction of decimals as “special fractions with denominators of 10 and powers of 10.” Four basic operations with decimals (which are similar to those of whole numbers). Whole number topics related to fractions, such as divisors, multiples, prime number, prime factors, highest common divisors, lowest common multiples, etc. Topics such as proper fractions, improper fractions, mixed numbers, reduction of a fraction, and finding common denominators. Addition, subtraction, multiplication, and division with fractions. 5 Tr. Belle used 2 1 4 instead of 1 3 4...
- Fuchang Liu(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...12 Fractions A Condition for Using Fractions: Equivalent Parts Jane told her children that they were going to learn fractions, and they were excited. To start with, she drew a circle on the board, then drew three vertical bars on it dividing the circle into four parts, and said, “A fraction expresses parts of a whole. We just divided this circle into four parts. If we take one part, that’s 1 of 4.” (see Figure 12.1). Figure 12.1 A Circle Divided Into 4 Unequal Parts Jane’s definition of a fraction, that it expresses parts of a whole, was correct. But this definition has a condition that needs to be satisfied: When we divide a whole thing into a number of parts, these parts have to be of equal size to be expressed with a fraction. Otherwise, the fraction may not be a correct representation of the parts in relation to the whole. In the figure Jane drew (Figure 12.1), the two pieces in the middle were larger than the two pieces on the outer sections. Figure 12.2 A Circle Divided Into 2 (a), 4 (b), and 8 (c) Equivalent Parts We may make an analogy to multiplication being referred to as repeated addition, in which case each set should consist of the same number of elements as any other set. Suppose Jane bought 8 packs of pencils and each pack contained 12. Jane could use multiplication to find out the total number of pencils she had bought: 8 × 12 = 96. However, suppose Jane bought 8 packs of pencils, but some packs contained 12 pencils each, some other packs contained 10 each, and still others contained 6 each. In this case she may not use 8 × ___ to solve this problem because not all packs contained the same number of pencils. A way to modify Jane’s picture so that a fraction may be used to express one or more parts of it is to draw lines that pass through the center of the circle...
