Fractions and Factors
Fractions are numbers that represent a part of a whole, expressed as one number divided by another. Factors are numbers that can be multiplied together to get a specific number. In mathematics, understanding fractions and factors is important for solving problems involving division, multiplication, and simplification of numbers.
7 Key excerpts on "Fractions and Factors"
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 - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...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. The category of rational numbers includes all numbers that can be expressed in the form a/b, where a and b are whole numbers and b is not zero. From the study of whole numbers, the conceptualisation of numbers falling on a number line that can stretch infinitely to accommodate increasingly larger numbers is promoted. Through the study of fractions and decimals, a further dimension to the number line occurs through locating fractions and decimals between whole numbers on the number line. The linking of fractions and decimals to per cent is further consolidated through locating percentages on a number line. Ratio, rate and proportion link to fractions, decimals and percentages, but understanding of these topics is not necessarily enhanced through the number line concept. Fractions The basis of fraction understanding is that fractions are a part of a whole—that is, when one whole thing is split into equal parts, each part is a fraction in relation to the original whole. Fractions can also be created from a set of objects numbering more than one. 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...
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 - ePubKnowing and Teaching Elementary Mathematics
Teachers' Understanding of Fundamental Mathematics in China and the United States
- Liping Ma(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...The teachers articulated it as “to find a factor that when multiplied by 1 2 makes 1 3 4”: As the inverse operation of multiplication, division is to find the number representing a factor when the product and the other factor are known. From this perspective, we can get a word problem like “Given that the product of 1 2 and another factor is 1 3 4, what is the other factor?” (Tr. M.) We know that the area of a rectangle is the product of length and width. Let’s say that the area of a rectangular board is 1 3 4 square meters, its width is 1 2 meters, what is its length? (Mr. A.) These teachers regarded the relationship between multiplication and division in a more abstract way. They ignored the particular meaning of the multiplicand and multiplier in multiplication and related models of division. Rather, they perceived the multiplicand and multiplier as two factors with the same status. 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...
eBook - ePubUnderstanding Mathematics for Young Children
A Guide for Teachers of Children 3-7
- Derek Haylock, Anne D Cockburn(Authors)
- 2017(Publication Date)
- SAGE Publications Ltd(Publisher)
...These represent the parts that you get when one whole thing or unit is divided equally into parts. The bottom number (denominator) indicates how many parts. The fraction three quarters (3 / 4) introduces a more general idea of a fraction by dividing a unit into 4 parts and taking 3 of them. This idea of a fraction can be extended to finding a fraction of a set, such as 1 / 3 of a set of 12 counters, or 3 / 4 of a class of 20 children. Suggestions for Further Reading Julia Anghileri has written extensively on the development of children’s understanding of multiplication and division. In a chapter entitled ‘Uses of counting in multiplication and division’, in Thompson (2008), she explores young children’s early experiences of these two operations. In a chapter entitled ‘Making sense of symbols’ in Anghileri (1995) she considers the meaning of multiplication and division and methods of teaching them to avoid later confusion. Anghileri (2006) also includes an interesting chapter on multiplication and division, combining the work of researchers with her personal knowledge of schooling. Chapter 10 of Haylock (2014) considers in detail the structures and associated language of multiplication and division. Chapter 15 provides a full explanation of what is involved in understanding fractions. We would refer the reader again to Nunes and Bryant (1996): Chapter 7 of this research into children’s understanding of mathematics deals with multiplication and division structures. Wright et al. (2006) provide strategies for assessing children’s understanding of multiplication and division, and suggestions on how to develop their understanding of these operations. Robert Reys and his colleagues (2015) have written a very comprehensive chapter on fractions in their book, Helping Children Learn Mathematics...
- 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...
