Fractions
Fractions represent a part of a whole and are written as a ratio of two numbers, with the top number (numerator) representing the part and the bottom number (denominator) representing the whole. They are used to express quantities that are not whole numbers, and can be added, subtracted, multiplied, and divided to perform mathematical operations involving parts of a whole.
8 Key excerpts on "Fractions"
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...
- 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 - 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 - 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)
...Their perspective, indeed, was legitimized by the commutative property of multiplication. The concept of Fractions as well as the operations with Fractions taught in China and the U.S. seem different. U.S. teachers tend to deal with “real” and “concrete” wholes (usually circular or rectangular shapes) and their Fractions. Although Chinese teachers also use these shapes when they introduce the concept of a fraction, when they teach operations with Fractions they tend to use “abstract” and “invisible” wholes (e.g., the length of a particular stretch of road, the length of time it takes to complete a task, the number of pages in a book). Meaning of Multiplication by a Fraction: The Important Piece in the Knowledge Package Through discussion of the meaning of division by Fractions, the teachers mentioned several concepts that they considered as pieces of the knowledge package related to the topic: the meaning of whole number multiplication, the concept of division as the inverse of multiplication, models of whole number division, the meaning of multiplication with Fractions, the concept of a fraction, the concept of a unit, etc. Figure 3.2 gives an outline of the relationships among these items. Figure 3.2 A knowledge package for understanding the meaning of division by Fractions. The learning of mathematical concepts is not a unidirectional journey. Even though the concept of division by Fractions is logically built on the previous learning of various concepts, it, in turn, plays a role in reinforcing and deepening that previous learning. For example, work on the meaning of division by Fractions will intensify previous concepts of rational number multiplication...
eBook - ePubGetting Parents on Board
Partnering to Increase Math and Literacy Achievement, K–5
- Alisa Hindin, Mary Mueller(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...Suggestions are made for using everyday Fractions, such as doubling and halving recipes or finding how many half-cup servings are in a three-cup bag of chocolate chips. See the bonus eResources for a sample letter. Putting It All Together Fractions are an abstract concept and thus difficult for many children to conceptualize; this is especially true for more concrete learners. Following the Common Core progression and integrating conceptualization of Fractions into our teaching is one way to ease the struggle of comprehending Fractions for students. The more children are exposed to the meaning of Fractions and real-life applications, the easier it will be for them to visualize Fractions and their relationships and use Fractions to solve problems. If we can convince parents to “talk” and use Fractions at home, then this exposure will be intensified. Children will begin to think of Fractions as numbers and operators, and use them to problem solve. When they think of Fractions as numbers, students will be able to visualize Fractions on a number line and compare them using multiple strategies that make sense to them. When they think of Fractions as operators, students will be able to think in terms of fractional parts of a whole and determine possible solutions....
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)
...By combining the knowledge in this chapter with an understanding of the language of math and a flexible interpretation of the symbolism, a student has multiple approaches for solving Fractions in different contexts. Making It Real. To focus on the concepts being discussed, the division examples up to this point have all had whole numbers as dividends for simplicity. Once students gain a solid understanding of these concepts, however, teachers can begin using more complex examples that include proper Fractions and mixed numbers as dividends. At the same time, students need to practice with real-life models or contexts. Repeated practice will enable students to take on tasks such as explaining the difference between dividing by 2 and dividing by, or the difference between multiplying by and dividing by. Box 8.34 lists some real-life problems that will help students build proficiency and understanding: Box 8.34 A. Quotative Model of Dividing by a Fraction 1. How many -foot lengths are in an object feet long? 2. Given apple per serving size, how many servings will apples provide? 3. A team of workers can construct miles of road each day. How many days will it take them to construct miles of road? B. Partitive Model of Dividing by a Fraction 4. A construction site has tons of bricks on hand, which is the amount needed for a job. What is the total amount of bricks needed for the job? 5. A train goes uphill from station A to station B and downhill from station B to A. The train takes hours to go from station B to A. This amount of time is the amount it takes to go from station A to B. How long does it take to go from station A to B? 6. A farm has acres planted in corn. The corn field is the size of the cotton field. How big is the cotton field? Conclusion Operations with Fractions, especially multiplication and division, can frustrate and bewilder students until their heads spin...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Fortunately, there are shortcuts in several of the fraction operations that make things easier. A fraction is made up of two parts, the numerator and the denominator : A fraction tells us how many parts of a whole thing. For example, the fraction means three parts of something that has four total parts. A mixed number has a whole number and a fraction, such as 2, which means 2 plus, and is spoken as “two and four-fifths.” The calculator available on the GED ® test has a function for Fractions that makes the calculations easier (and more accurate). However, understanding Fractions is essential for using the calculator to get the right answers. The reason Fractions look like division problems is because that is actually what they are. We already saw how to get a decimal answer by dividing two numbers, so we just have to remember which part of the fraction gets divided into which. C ALCULATOR Fractions The important keys for calculating Fractions are, scrolling, the toggle key, and. To enter a fraction, enter the numerator, press, then enter the denominator. The screen will look like. Scroll to the right to leave the fraction mode. For example, for, press 3,, then 4, scroll right. Use the toggle key, then to display the fraction as a decimal. Note that if the fraction has more than a single entry in either position, enclose the expression in parentheses. For example, to evaluate, press, then 4, scroll right. To enter a mixed number, use, then. The screen will look like. Enter the whole number part, scroll right, then enter the fraction part as above. For example, for 2, press, then, then 2, scroll right, and continue for as above, but in this case scroll after the numerator (3) to enter the denominator. To convert any fraction to a decimal, just use the toggle key, and. To convert an improper fraction to a mixed fraction, use, then. For example, to get the mixed fraction for, press 21, then, then 5, scroll right, then, then...
